Let's open our Bibles to 1 Samuel chapter 2. We're going to be looking this morning at verses 1 through 11 as we have just begun a study here in 1 Samuel. And we'll be looking at Hannah's prayer this morning here in chapter 2 of 1 Samuel. Tonight we have our evening service and we obviously have our midweek on Wednesday and like I was sharing with you last week it would be good if you were to choose a place to come for an evening service if you're able and plug in. And so we do have our evening service, we're in the book of Ruth and then on Wednesday we continue our study in the Gospel of Luke and so I'd love to have you with us if you're able. I might mention the ministry fair, you can see what that is in a couple of weeks. See kinds of ministries that we have here and where you might want to plug yourself in. Super Bowl is coming up in the Super Bowl weekend and so we have our men's breakfast. We're going to have Miles McPherson with us and some of you have been with us as Miles has been out here with us more than once and he's always a blessing and he pastors a church in the San Diego area. They have five Sunday morning services and he has a 3,500 seat sanctuary. The Lord has just used him tremendously and it's always a blessing when Miles comes out and shares with us and so I'd invite you to be part of that fellas when that takes place at the end of the month. And then finally we do have our couple's banquet that's coming up in February on the 13th and that's for dating as well as married couples and if you'd like to be part of that of course we'd love to have you with us. Today we're in chapter 2 here in First Samuel. We'll read verses 1 through 11 as we continue our study that we just began in the book of First Samuel and we're going to be looking at a section that I chose just to call Hannah's prayer. It's going to be obvious as to why I called it that in just a moment because it says in First Samuel chapter 2 verse 1 and Hannah prayed and said, my heart rejoices in the Lord. My horn is exalted in the Lord. I smile at my enemies because I rejoice in your salvation. No one is holy like the Lord. There is none besides you. There is nor is there any rock like our God. Talk no more so very proudly. Let no arrogance come from your mouth. For the Lord is the God of knowledge and by him actions are weighed. The bows of the mighty men are broken and those who stumbled are girded with strength. Those who are full have hired themselves out for bread and the hungry have ceased to hunger. Even the barren has born seven and she who has many children has become feeble. The Lord kills and makes alive. He brings down to the grave and brings up. The Lord makes poor and makes rich. He brings low and lifts up. He raises the poor from the dust and lifts the beggar from the ash heap to set them among princes and make them inherit the throne of glory. For the pillars of the earth are the Lord's and he has set the world upon them. He will guard the feet of his saints but the wicked shall be silent in darkness for by strength no man shall prevail. The adversaries of the Lord shall be broken in pieces. From heaven he will thunder against them. The Lord will judge the ends of the earth. He will give strength to his king and exalt the horn of his anointed. Then Alcona went to his house at Ramah but the child ministered to the Lord before Eli, the priest. As we have been looking at First Family, we know that Hannah had been barren, married to a man by the name of Alcona. Alcona had a second wife, a woman by the name of Panina. And Panina was able to have children and because Panina was having children and Hannah did not have any, Panina would mock and would cause great pain to Hannah because Panina would let her know that she felt herself to be favored even though Hannah was the one who truly was favored by the husband Alcona. It caused her great pain and so Hannah had taken this request to the Lord. She had actually prayed in bitterness of spirit and had spoken to God and had said to the Lord, God I want you to give me an ability to have a child and in chapter one at verse 11 she had said to him if he was to give her a child I will give him to the Lord all the days of his life and no razor shall come upon his head. She had said Lord if you give me the ability to have this child I will dedicate him to your service for all of his life and so she had gone to the tabernacle. The priest Eli was there and Eli watched her as she was praying and could see her lips were moving but no sound was coming from her mouth and therefore he thought that she was drunk and he approached her and he said to her she should put away her wine.  She immediately says listen it's not because I'm drunk that I'm here I'm here praying I'm a woman of sorrowful heart because I desire to have a child I'm barren I'm unable to have one.  So with joy she goes home and as she goes home over a period of time she and Alcona have normal relationship she becomes pregnant conceives this child and is greatly excited about it and so she says to her husband I'm not going to be going up to the tabernacle until I've weaned Samuel and so we know that in the time of Samuel that the mama would continue nursing the baby up to three years and so now after three years have passed she has taken this child to dedicate him to the Lord and so in verse 2028 of chapter one she says there I have lent him to the Lord as long as he lives he shall be lent to the Lord and they worship the Lord there and so he's three years old she's dropping him off at the tabernacle he's about to begin his service to the Lord and she's going to leave him there for the rest of his life and the question has to be asked how does she feel how does she feel about this what's going on in her heart you see she has taken this request and she gave it to the Lord she had prayed to God she had asked him for a son and she said I will offer him to you for service she said he will be a servant he will live a holy life all the days of his life and therefore she came to God the God who was able and she made a request known to him she knew that this God was not like the gods of the pagan she knew that this God was not the God of the heathen this is not a distant and incapable of hearing type God this is a God who's able to minister he is with us he's near to us and all she needs to do is make her request made known to him because she knew that prayer was a conversation that she was having with her God and that's what prayer is it's a conversation that we have with God it's a subject of a lot of Bible I mean when you start in Genesis go to Revelation there are over 350 distinct prayers that are recorded for us in scripture you see that prayer is a heartbeat of a believer and there are times that you speak to the Lord in a variety of ways there are times that you might be praying to God in a way that you're simply giving him thanks for the good things that he's done there are times that you're giving to the Lord in your in your prayer just a lot of praise for the goodness that he has has been to you how good he's been there are times that you're you're lifting up your supplications to God where you're you're making a request known to him something personal something you desire there are times that you're interceding you're speaking for somebody else you're saying God in Jesus name I'm lifting up this person to you there are times that when you're praying you're actually making a complaint to him you're saying Lord I just have to to let you know how I feel about what's going on and then there are other times that you might be simply asking him a series of questions it's prayer it's conversation with God and it's something that Hannah was very familiar with and she would bring these requests to the Lord and she did so because she knew that God would answer you see God actually in Scripture gives to us an invitation to approach him in Jeremiah in chapter 33 verse 3 he said call unto me and I will answer you and and show you great and mighty things which which you know not God actually gives to us an invitation to approach his throne and to bring our request to him he invites us to do that now before I was saved I might pray because people do but I never had a confidence that he was listening and didn't know that he actually is listening to me I always sense that there was some kind of separation there was something that was dividing us and I didn't know what it was but as I got saved I came to realize what it was and what was dividing what was making separation was my sin Isaiah 59 verses 1 and 2 says behold the Lord's hand is not shortened that it cannot save nor his ear heavy that it cannot hear but your iniquities have separated you from your God your sins have hidden his face from you so that he will not hear there was a sin separation something that was making my conversation with God was blocking it and and when I got saved that that sin separation was dealt with and and because of the Lord Jesus Christ now I can come to him with a full confidence that he's listening to me I can come to the Lord and and speak to him and and know that he listens because I have been forgiven by God Ephesians 2 13 says in Christ Jesus you who once were far off have been brought near by the blood of Christ and so I can now approach him I can come to the Lord and speak to him Hebrews 10 19 says we now have boldness to enter the holiest by the blood of Jesus so as Christians we know that God hears this because we know that God loves us access has become ours we can come to him as his children whenever we need to speak to him the psalmist tells us in Psalm 34 15 the eyes of the Lord are on the righteous his ears are open unto their cry and so we can come boldly to the throne of grace obtaining mercy and find help in our time of need because God has made that way available to us we can pray and that's what Hannah is doing now sometimes when you're praying you can have the sense that maybe maybe he's not going to answer maybe there's something that I have to deal with you know I've discovered that sometimes when I pray my prayer may be just an extremely selfish prayer it's just selfish now there's this program that's coming on real soon some of you have seen it others won't admit to it that you've watched it but it's called American Idol American Idol a lot of people watch it especially the beginning programs when you see these people come up who cannot sing and and they they try so hard and and you sit there guiltily kind of laughing saying oh God I feel so guilty but this I can't believe this and so you watch it as this person's out there trying to sing but don't you think that that person before they went up to do their their song don't you think they were praying and saying God I'd like to be an American Idol they are they're saying God help me to win and I've discovered that sometimes the Lord says no you can't sing I gave up praying when I was watching my favorite team play a long time ago because he never answers yes Lord let you see LA beat the Trojans just this one time he says no no no no no doesn't work you know so I gave up a long time ago I don't pray for athletic contests I just watch and suffer there are some things that you can ask for that the Lord is not even going to dignify with an answer sometimes we're asking because we're asking incorrectly it's kind of like what James says in chapter 4 verse 3 when he says you ask and do not receive because you ask a miss that you may spend it on your lust you're asking for the wrong thing you see according to 1st John chapter 5 verses 14 and 15 we should pray according to the will of God and if we know that we're praying according to his will we also know that we have the petitions we've requested of him when you ask according to his will well sometimes we may pray but we don't even expect God to answer we're not expecting him to respond to us at all it's kind of like we just throw the prayer up in the air hoping that somebody up there might be listening but the Bible tells us we're to ask in faith without doubting for he who doubts is like a wave of the sea driven and tossed by the wind a double minded man is unstable in all his ways you can't be asking for one thing with no intent to actually receive it you pray with faith but sometimes we don't sometimes the Lord just doesn't give us the answer simply because it doesn't line up with his will for us so so we need to know that God does answer prayer a God will answer yes God may answer no and then sometimes God answers wait but we lift our requests and we ask the Lord if he might move and we do so because we know that God is good and we know that God loves us so that's what had happened with Hannah Hannah had taken her broken heart to the Lord God and she had made her request of him and as she had said as she had made her prayer to him the Lord had stated that he would he would answer through Eli and she was able to go home and so as she's gone home she has received the answer of the Lord she finally brought the baby and dedicated the baby and she's leaving him there with them as it says in verse 28 chapter 1 again I have lent him to the Lord as long as he lives he'll be lent to the Lord and they worship the Lord there and so now Hannah begins to pray and in chapter 2 verse 1 we have a chance to see what she prays is she upset is she discouraged is she sad she's leaving her three-year-old behind she's about to leave and go home without the joy of her heart this is a three-year-old baby that has been everything to her the answer to her prayer and she's going to leave this baby there with Eli and she's going to leave him and go home by herself she's been with them for three years she's been nursing him loving him speaking to him training him devoting herself to him by the time he's three years of age he's already learning prayers he's able to communicate there are things about this three-year-old that must have just grabbed her heart she loves him with everything that's within him within her but what's going to go on how does she respond well notice what she says in verse 1 here Hannah prayed and said my heart rejoices in the Lord my horn is exalted in the Lord I smile at my enemies because I rejoice in your salvation Hannah begins by saying my heart rejoices in the Lord when she says my heart that speaks of her center of moral and spiritual life she says everything within me is rejoicing it's not a time of sadness it's not a sad time of letting go it's a moment of triumphant rejoicing God has answered my prayer I am overjoyed she says my horn is exalted in the Lord I smile at my enemies because I rejoice in your salvation when she says my horn is exalted that word horn in the Old Testament refers to various things it can include my pride it can include military it can include political power normally it speaks of strength because an ox is a very powerful animal and they would use the horn as a symbol of the strength and so she's speaking about the strength that she has and she's saying I have an inner strength that has been given to me it's been produced by God's goodness as he strengthens me and as a result of that I smile at my enemies I smile at them because I rejoice in your salvation your salvation has caused any sorrow that I could have within my heart to be turned to joy and this inner joy that you have given to me causes me to smile that which is on the inside has found a place of expression on the outside my heart is smiling and so is my mouth and I have come to worship you and I've come to give praise to you because you have been good to me I rejoice in your salvation Isaiah 61 verse 3 God promises there to console those who mourn in Zion to give them beauty for ashes the oil of joy for mourning the garment of praise for the spirit of heaviness and so she's saying you have given to me the joy of my heart and the great desire of my heart and I smile at those who at one time have been oppressing me she goes on in verse 2 and she says no one is holy like the Lord there's none besides you nor is there any rock like our God this is my joy these are the reasons for my joy I am joyful because you are holy I am joyful because you are without equal in the multitudes of the so-called gods there is only one God and that God is my God there is no rock like our God when she says nor is there any rock like our God a rock is immovable it's enduring it gives a picture of strength and it gives to us an insight that God is there to help us when we're in our times of need she's rejoicing because the Lord of might the God of all strength has heard her cry and he has listened to her he is her God and he is her stability so she speaks now and verse 3 is interesting because she's undoubtedly addressing this really to what she's gone through with Panina talk no more so very proudly let no arrogance come from your mouth for the Lord is the God of knowledge and by him actions are weighed now notice the Lord is the God of knowledge when it says the Lord is the God of knowledge God has every kind of knowledge because God is omniscient God knows everything like the psalmist said in Psalm 4421 he knows the secrets of the heart or Psalm 147.5 greatest our Lord mighty in power his understanding is infinite God knows all things is what she's saying and he's able to judge righteously God has seen what you have done Panina how arrogant and proudful you have been towards me and God has heard my cry God is the only one who is able to righteously judge you know sometimes somebody on the outside you look at him and you'll say man that's a good guy or that sure is a beautiful person there because on the outside they look that way they use good words they can sometimes even speak religiously and you may look at them and all on the outside they look like they're a very righteous person and then there are other people that you see them from the outside and you would think the opposite of them you would think that this is a rough person this is an ungodly person I have met some guys who are huge I mean they're just huge guys who have such gentle and loving hearts they love the Lord and they would give you the shirt off your back so you could make a tent out of it I mean they're so big they're just great guys you know but if you look at them at first you know you're thinking oh this is but it turns out no you know outer appearances can fool you we don't have the ability to judge a person's heart and that's the point she's making she's saying God does God has all knowledge God knows what's inside of them when the hippies were getting saved and we would share our faith with people who were not hippies all they could see was the long hair and the bare feet and they didn't want to hear the gospel they couldn't see our heart all they could see was the outer appearance today when kids get saved and they've got tattoos all the way down to their fingernails and they've got piercings and everything a lot of times all the people can see is the piercings all they can see is the tattoos they don't see the heart they can't see the heart but that heart is right with God and that heart's a good heart towards the things of the Lord but because people have a tendency of judging on the outer appearance well we need to understand that God sees the heart and so that's what she's saying here let no arrogance come from your mouth the Lord is the God of knowledge by him actions are weighed God sees all things she says in verse 4 the bows of the mighty men are broken and those who stumbled are girded with strength so she begins to speak how God works she begins by saying the mighty are weakened but the weak are made mighty the proud can be reduced to a place of humility but the humble are lifted to prominence Proverbs 3.34 says surely he scorns the scornful but gives grace to the humble so God works the humility secondly she says in verse 5 those who are full have hired themselves out for bread and the hungry have ceased to hunger and so she goes on to say those who are rich or full that would speak of the landowners the landowners who are working in their own fields and the servants who are now at rest so she's saying God has a way of taking the powerful and reducing them and the weak and elevating them she says in verse 5 even the barren has born seven and she who has many children has become feeble God has made it made it possible for the barren to be perfectly happy the number seven in scripture is a sign of perfection or completeness and so it's a picture of somebody being perfectly happy and so she's saying even a person who didn't have children can be made perfectly happy in the Lord and even have many children it's interesting how that she had been barren and had Samuel but Samuel's not the only one that she speaks of notice in chapter 2 here verse 21 notice what it says in chapter 2 verse 21 how it says the Lord visited Hannah so that she conceived and bore three sons and two daughters meanwhile the child Samuel grew before the Lord she went on to have six children altogether not just the one that she had but she had five more and so she's saying God has made it possible for me to be perfectly happy in him she goes on in verse 6 to say the Lord kills and makes alive he brings down to the grave and brings up it is the Lord she is saying who gives life only he has the ability to give life and only he is the one who takes it away when you read the book of Job in chapter 1 verse 21 Job said it this way Job said naked I came from my mother's womb naked shall I return there the Lord gave the Lord has taken away blessed be the name of the Lord God is all powerful and he is the God of life she says in verse 7 the Lord makes poor and makes rich he brings low and lifts up and so God has a way of putting into our into our hands and ability to gain riches and sometimes he even removes riches from us our financial situation often rests on our relationship with the Lord and so she says God is able to make someone rich and somebody else poor but it's God who is the one who is in charge of all of that verse 8 he raises a poor from the dust and lifts the beggar from the ash heap to set them among princes and make them inherit the throne of glory God can put you in a place that you have no intrinsic rights or abilities to inhabit God is able to do that when I consider that I think I have a bit of understanding in that one verse there he lifts the beggar from the ash heap to set them among princes God has been extremely good to me and as I look in this room I believe that God has been extremely good to all of us you know God has a way of working things out to put you in a position that you may realize you're not qualified for and yet he places you in that position you may be somebody working in an office or in a warehouse or wherever it may be and you started in a certain position but now you are a supervisor you're overseeing things you may be being moved up into the company ladder and they're giving you more and more and yet you're thinking within yourself I don't deserve this because I didn't graduate from college didn't even go to college and yet you may be saying and yet look where I'm at I'm overseeing responsibility and there are guys here with master's degrees or even doctorates that I have an overseer relationship with and you feel so humbled because you're put in that position because you have some experience you gained you gained it in life perhaps you served in the military gained some experience there but you are now a supervisor in a place that is filled with guys who have upper degrees and yet you don't have that and you look at it and you think Lord you're too much how did you put me in this position how did I get here how did you place me here it's because God has a way of taking you from one place and putting you somewhere else God has a way of doing that God has a way of taking you and elevating you and putting you in a position that he wants you to be regardless of whether or not you feel the ability to do that regardless of whether you have the qualifications or the education God has a way of doing that and he places you in that position and it blows your mind when he does and you see what you are doing and how God has been good to you and it just amazes you at the goodness that God has shown to you I understand that I remember getting an invitation several years ago the first one I received I received more than this but the first one I received I remember getting a call saying you've been invited to come to the White House but you have to send some your driver's license information and all pertinent information that relates to you you have to send it to the FBI they're going to have to screen you to see whether or not they're going to allow you in or not and I remember Marie and I having to jump through all those hoops and finally they sent us an invitation where to meet there in a small meeting with the President of the United States and how that we jumped on a plane flew to DC went to the hotel the next day you know got dressed went you go to this little gate you give them your license you give them your name they run you through they send you through this one gate you go into another room they interview you you go through another gate go through a check see if you're carrying any weapons send you to a little elevator go into the elevator and you go down in this room and you go in and you go and you find seating in this small room with about seats about a hundred people and you're in there and it's not full and you're just sitting there and a few just a few rows from the podium from the platform there in a small small room and then as you're seated there you hear the music and then you hear somebody say ladies and gentlemen the President of the United States and and everybody stands as he walks in and you're you're within ten feet of this man that you see on TV all the time and you know who he is and I turned to Marie and I look at her and I say some of you will understand this some of you may not but I looked at her and I said who would think the first time I was ever there who would think a little Mexican guy from our walk in the same room with the President of the United States I was humbled I was humbled some people though I hate him just the idea of being there just the idea that I was there and I've been invited there more than once you're humbled you're humbled I graduated from high school with a D minus average that scares you huh I cut classes in my senior year for the whole year I showed up only for tests and once in a while just so they had me on the rolls I was never in class if they had me doing the detention that I should have been doing because I used to give you an hour's detention if you cut a class I'd still be making up all the hours 41 years later but God has a way of taking you from the ash heat and putting you among princes He has a way of doing that I was at a wedding two weeks ago and as I was seated at the table a woman approached me and walks up to me and looks at me and she says to me excuse me are you David Davros Hollis I said yes I am who wants to know how much do I owe you no yes I am she goes I went to high school with you really what's your name she gives me her name I knew her from high school 41 years ago I haven't seen her in 41 years and as she's talking to me she goes we went to high school together and I said I remember you of course she says now who did you hang around with I said every troublemaker in high school was my friend I hung around with that group she says like and I started naming all the troublemakers I hung around with I hung around this guy that guy oh yeah oh yeah oh yeah we were we were troublemakers in the school we were the drugies I mean we were the ones who would go to lunch get loaded come walking in with sunglasses on and just sit there and crash for an hour that was us we were the ones who go out drinking during lunch time we were the ones we were the trouble in the school my friends and I caused all the problems we were the ones who made Sierra High School class of 1968 the number one drug school in the Whittier Union High School District it was us and when it was announced that Sierra High School is the number one drug abuse in high school we cheered yeah that was us she remembers me I said all those guys those were my friends and you're a pastor now? I'm amazed I'm amazed at what God has done and I said you need to come and visit us perhaps you will come and visit us she lives in the area come and see what God has done see what the Lord has done I've had people come to church here it's happened twice now where they have two different guys with different friends who have come and they said I knew him from high school he can't have changed like that it's not possible that David's doing this today but I have because of what the Lord has done because God has a way of taking the beggar from the ash heap and sets them among princes God has a way of doing that God has a way of doing amazing things God has a way of transforming your life God has a way of putting you in places you don't deserve to be respected with an amount of respect that you really are not worthy of but God has a way of doing that he does have a way of placing you in places that you couldn't have opened the door but God opened the door God placed you in God said you be seated here because you are somebody I'm delighting in and I am amazed I'm amazed at the goodness that God has shown me I am amazed because I'm married above my pay grade I know that I have four children that I adore and two grand babies that are the heartbeat of my life God has blessed me with friends and family and a church that I love with all of my heart he is a God who gives you such great blessings and grace and benefits and goodness he has a way of taking you and all you need to do is begin to look at where you were and where you are now to see the hand of God what God can do what God can do look at where you were and look at where you are and look what God has done for me 38 years of walking with Jesus Christ and I have seen God do amazing things that I do not deserve and God is the one who does it he has put me in places where I have opportunities you know I've traveled internationally to go to pastors conferences to teach pastors in Austria to speak to them in various countries of the world Mexico and you name it I've had opportunity to go to South America I've had opportunity to go to Israel more times and I can even remember to do ministry in so many places to speak in front of thousands of people like at the Anaheim Men's Conference and men's conferences throughout the nation I have seen God and I am amazed by it and there are times when I just am overwhelmed by it like you I see God's hand on me and what God is doing and I am amazed and you ought to do the same you ought to be looking back and saying God you have been so good to me look at what you've done look how good you've been you have done so much and the thing is he's able to do abundantly above all we could ask or even think and so Hannah's praying and praising God and rejoicing in him the one who was barren has been given seven children in other words she is saying I am perfectly happy because I have seen what you have done God you've taken the humble and placed him in a place where they can be used and those who thought they were to be used you've humbled them look at what you have done you are the God who does this she says in verse 8 the pillars of the earth are the lords he has set the world upon him God you're the one who keeps all things stable you keep this world from crashing in and of itself and upon itself even as Psalm 102-25 says of old you lay the foundation of the earth the heavens are the works of your hands and we know according to Hebrews 1-3 that Jesus upholds all things by the word of his power you are the one who keeps all things together you are the one who keeps things solid he will guard the feet of his saints but the wicked shall be silent in darkness for by strength no man shall prevail God's people follow him and he leads them in security and righteousness he protects them every step of the way even as it says in Psalm 37-23 the steps of a good man are ordered by the Lord and he delights in his way God takes care of you and he leads you in the path that he would have you to go paths of peace and paths of righteousness but the unrighteous live in darkness and they stumble and ultimately are left in darkness Matthew 25-30 Jesus has cast the unprofitable servant into outer darkness there will be weeping and gnashing of teeth you see because by strength no man shall prevail a man or a woman is made strong when they give up when they yield when they say enough I'm sorry God forgive me by strength I cannot prevail by my own effort by my own power I can never accomplish anything for God it's only what I do by the power of God's Spirit that ever counts for anything I cannot prevail I cannot do I cannot save myself I cannot be victorious in life neither can you but when you walk with the Lord God opens doors no man can close and God places you in positions to be used and because your heart is right with him God is glorified through you but when you're walking outside of the Lord you walk in darkness and ultimately that's where you remain he says in verse 10 she says the adversaries of the Lord shall be broken in pieces from heaven he will thunder against them the Lord will judge the ends of the earth he will give strength to his king and exalt the horn of his anointed ultimately the enemies of the Lord the adversaries will be broken this is a prophecy because she's speaking concerning and I want you to see this in the last portion of chapter 2 verse 10 where it says he will give strength to his king and exalt the horn of his anointed that word anointed there the horn of his anointed that word anointed is speaks of Messiah the anointed is Messiah in the Hebrew it speaks of Messiah and she's saying ultimately victory is going to come this is a prophetic word ultimately victory comes through Messiah Messiah is bringing victory the adversaries will be crushed and all who oppose Messiah Jesus will be overwhelmed under his anointed all enemies will be vanquished in Psalm 2 verses 6 through 9 the Psalmist said I have set my king on my holy hill in Zion I will declare the decree the Lord has said to me you are my son today I have begotten you ask of me I will give you the nations for your inheritance the ends of the earth for your possession you shall break them with a rod of iron you shall dash them to pieces like a potter's vessel Jesus Christ ultimately is going to be victorious and she's prophesying concerning you Messiah who ultimately will come and deliver in the nation of Israel as well as all who believe in him and so she's praying she's simply saying this she's saying I came to God with a broken heart I had an adversary by the name of Panina Alkana my husband's second wife she was jealous concerning me because obviously Alkana loved me I couldn't produce children for him so he married a second woman she produced children and used those children as a weapon against me it broke my heart it got me to the point where I was inconsolable but I took my need to the Lord and though seemed like an impossibility I asked God would you please make it possible for me to have a child and God granted my petition but I had said to God I will give this child to you to serve you in holiness all the days of his life and I have come in order that I might dedicate Samuel to service because God has heard my request I don't walk away from this place crying I walk away from this place rejoicing because God has given to me the desire of my heart and I asked him if he would and he did and as he has done so I dedicate this child to service to God and this child shall serve him all the days that he lives and it says in verse 11 then Alkana went to his house at Rama but the child ministered to the Lord before Eli the priest and so she leaves him there at that tabernacle she leaves Samuel there to the work of the Lord and as she leaves she leaves him thinking that he's in a location in a place where he's going to learn things that will make him a more righteous child but the sad fact is that even in that tabernacle even as this child is dedicated to serving God even in this holy environment unholiness is rampant and we're going to be seeing that next time we get together even in the midst of a place that is dedicated to God there is unholiness that is finding places of expression we'll see that next time Father I ask that you would work in us I ask that you would continue to show us Lord your ways that you would work within each one of us as we yield ourselves to you Lord we see that you are a God who answers prayer there have been times that we have prayed Lord and haven't received from you because we have asked to miss desiring to consume it on our own lust we ask for things Lord that really weren't of your will or we ask Lord sometimes not really sincerely expecting you to move at all we know that you answer prayer we know that you answer yes you answer no sometimes you answer wait but you do answer prayer Lord and so we would learn to pray too we would learn to put our concerns and our cares upon you we would learn that you are a God who is holy and just and you are a God who elevates those Lord who humbly walk with you Lord I want to thank you for the goodness you've shown to us to this fellowship to many in this room how you have put us in positions of influence how you have worked in our lives and glorified yourself how you transform people who yield to you and I want to lift up this congregation Lord that we might learn to pray knowing that you are a God of all power and all knowledge you already know our situations and you have the ability Lord to deliver us we would lift our concerns and conditions to you even right now some in this room Lord need a touch because their bodies are breaking down some in this room need a touch from you Lord because they're having difficulty making home payments Lord or because Lord they're in danger of losing their job or they already have some are concerned for children Lord some are concerned for their marriages some are concerned for their parents many are carrying burdens and hurts right now that only you are really fully aware of Lord so we lift these requests to you now Lord and I pray that you would reach down and show them that you are the God who is able you are the God who cares and may they walk out of this room trusting you Lord as they cast their cares on you even now our eyes are closed and our heads are bowed perhaps I have some in this room right now we need prayer you need to get right with the Lord you need to yield to him right now I want to pray for you as our eyes are closed and our heads are bowed would you raise your hand let me pray for you right where you're at just raise it so I can see you please Lord you see these hands you know the reason why they're being raised to you I'm asking that you would reach down now and touch them represents needs Lord these your children have I pray Father that you would bring comfort in that you would cause them even now to know you love them you're concerned for them and you're able to meet these needs Lord and I ask that as they raise their hand to you that you would raise them Lord that they might walk with joy knowing that you are in control and they can trust you even as as Hannah walked away rejoicing may they walk away knowing that they've cast their care on you and you care for them wash them and cleanse them Lord we're needed fill them with your spirit and work within them Lord that they might from this day forward just enjoy you and trust you in all of this and we thank you Lord and bless you thank you you can put your hands down Jesus I ask that you keep moving in all of us that we would learn how to pray and learn who you are as we pray and I ask this now in your name Amen let's all stand we'll close with a song and a word of prayer our father we ask that you work in us and through us and that you use us for your glory we leave this place into the mission field we will serve you and may we be found faithful as we do so draw us back tonight Wednesday whenever that we might continue fellowshiping Lord and growing in you and we ask this in Jesus' name Amen God bless you


So, so the host of condition is the following, no? We have two different points. Any two different points have these joint neighborhoods. That's host of, or T2, okay? And T1 is, any two different points have, each one has a neighborhood, which doesn't contain the other one. That's T1. So these are two separation axioms, okay? There's also T0 and we will see T3, T4. They are not the same. So for example, if X is, X has a final complement topology, a set with final complement topology, is T1. This is T1. Why is it T1? Well, if you have any point here, okay? If you have any two points here, no? You want the neighborhood of this, which does not contain this. And what is the neighborhood? That's the whole space minus the other point, okay? A neighbor in the final complement topology, final complement. You take the whole space minus this point. That's an open set, okay? Which contains this point, obviously, right? Final complement topology. X minus one point is open, okay? So X minus this point is open and contains this point. So this is T1. Each point has a neighborhood which does not contain the other point, okay? So it's T1. If X is infinite, it's not T2. But not T2. If X is infinite, if X is finite, and has a final complement topology, what is the topology, another name? We have a finite set and we have the finite complement topology. There's no condition, right? It's a discrete topology, okay? There's this house stuff, obviously, no? The discrete topology is house stuff. The points are open, right? So this is again, if you want, so this is an exercise, okay? So you have two points. Suppose two points, two distinct points have, I want to prove this, okay? Suppose two distinct points have disjoint neighborhoods, okay? Any two distinct points, suppose, we have the finite complement topology, okay? Suppose X and Y have disjoint neighborhoods. So this is an exercise, U1, U2. So this would be X1, X2, no, X, well, X1, X2, okay? So it doesn't matter, but let me call it X1, X2, U1, U2, okay? X1, X2, and this is U1, this is U2, disjoint neighborhoods, so we are in this situation, right? This means U1 intersection U2 is empty, they are disjoint. That means X is equal to X minus U1 intersection U2. This is empty, no? And we have again the Morgan, always the same, in set theory, almost. X minus U1, union now, okay? So it goes to the union. X minus U2, right? This is the Morgan again, but these two are finite. We have the finite complement apologies. These are neighborhoods, U1 and U2, means that the complements are finite. So this means that X is finite. Then we have the discrete apology in this house of course, okay? So if X is infinite, it's not house of, right? Yesterday he made the remark, somebody of you made the remark. So what is the definition, let's give the definition of, we have to give that anyway at a certain point, okay? And you know that, of convergence. So X is a topological space and we have a sequence. A sequence, X1, X2, X3 in X converges to X in X to a point. X in X now, so what is the definition? If for every neighborhood U of X, there exists capital N, a natural number, there exists capital N in N, so a natural number, such that, such as what? XN is in U for all large N. So XN is in U for all N bigger equal to this fixed capital N. This is the notion of convergence, okay? That's the definition of convergence in general topological spaces. So we write, we will write XN converges to X, okay? If this, okay, that's a convenient way. XN converges to X and what you remarked, somebody of you remarked yesterday, so in a house of space, if a sequence converges, then the limit point is unique, okay? So observation, that's trivial almost now. If XN converges, then the limit point, the limit X is unique. So if you have this point X, this is the sequence XN, and this is unique, it cannot convert because if you have any other point, Y, okay? Then we have disjoint neighborhoods, sorry. If it is house of, if X is, that's the most important, which I forgot, I'm talking about house of spaces, no? If X converges and the limit point is unique, if X is house of, that's the most important here. If it's not, one second. So if it's house of, then for any two points, we have disjoint neighborhoods, okay? And then for large N, this sequence is in the neighborhood U of X. This is small X, okay, in this neighborhood. So it's not in this neighborhood. All these points are not in this neighborhood, which I just wrote, okay? So in a house of space, a convergent sequence converges against a unique point to a unique point, okay? If it's not house of, it's not true.  In the indiscreet topology, every sequence converges to every point. There's only one neighbor, which is the whole space, and the sequence is in the whole space, so that's a very stupid suggestion, no? In the discreet, in the indiscreet space, indiscreet space, every sequence converges to every point. In the discreet space, what are the convergent sequences? Discreet, no, no, what? Discreet, discrete, what? Yeah, what are the convergent sequences? Not exactly, what did you say? The convergent sequences are constant, almost. Not exactly, they are finally constant. So the first is not important, but up a certain point, they have to be constant. Not constant, but you say finally constant. At a certain point, they become constant. Before it's not important, okay? The first terms are not important, right? So not constant, but finally constant, okay?  We have, let's say, the exercises are more pleasant than the theory sometimes. We have, let's say N with a final complement topology. The natural numbers with the final, the natural topology with the natural numbers is a discrete topology, okay? The subspace of the reals, it's discrete, okay? The natural. Now we take the final complement topology, which is one of our examples, okay? The final complement topology, this is T1, no? It's T1, not T2, no? Not T2, T1, not T2. Not house stuff, that's what we, that's stuff. So then take the sequence one, two, three, four, five, and so on, no? Does it converge? And if yes, what is the limit, huh? It converges against one, is that true? Yes, it's true, it converges against one. Any neighborhood of one has final complement, so excluding finitely many, you have everything in this neighborhood, right? Every neighborhood of one is open, has final complement, so it contains all natural numbers, except finitely many, which you can forget, okay? So up a certain point, everything is in this neighborhood, okay, all natural numbers are in this neighborhood, right? So it converges against one, and then, other terms, it's not what, it converges against one, and there are other points? Everything, it converges against every point, right? Yes, right, one, two, three, so if points, okay? It converges against any point, no?  But that's the case. This is T1, so T, I said yesterday, in this course, later, we always use T1 almost, okay? T1 is sufficient very often, no? But for converging, it's not sufficient, no? So it goes very wrong with T1, no, the unique limit, no? T2, it's unique, okay? But in T1, we can have everything, okay? Well, maybe your question will talk later, okay? Because I have to go on with this theory, I suppose. So why did we introduce this at this point? So there's a proposition, okay? Let me write this proposition, so proposition. So extra Hausdorff space, extra Hausdorff space, you never say T2, you say Hausdorff, that's it, okay? T2 is too technical. And then somebody forgets, what is T1, what is T2, no, Hausdorff is clear. And A subset of X and X and X, so this is the situation. X is the limit point of X, so it's the same. X and X is the limit point of X, of A, sorry. It's the limit point of A, if every neighborhood of X, if and only if, if and only if, if and only if, so I'll write this, if and only if, every neighborhood of X intersects A in a point different from X, that's the definition, no? And now we have a Hausdorff space, we say it intersects every neighborhood, abbreviate, neighborhood of X intersects in a point different from X, it's the definition of limit point, no? And now we saw in infinitely many points, okay? So it looks a little bit more of a limit, but we still have no limit, okay? But in infinitely many points. So we come a little bit closer to limit, no? At least you find infinitely many points. But limit means we have a limit, but that's still not true, okay? There we need another condition, a stronger condition, which we will see, okay? No limits, but infinitely many points. Well, okay, so sometimes I give no proofs, now I give two proofs because they are both easy and nice. So the first one, so here are two proofs. First proof is constructive, no? You make that picture. Well, let's make a picture. So we have X, we have A, and we have, this is A, this is X, and here we have our point X, okay? And now, well, one direction is trivial, no? Which one is trivial? If you have this, then X is the limit point, okay? So that you, I forgot to write, this is trivial, okay? So this direction, no, which one, the other one, no? Yes, this is trivial. So we prove the other direction, okay? So let you be enabled of X. It's, this follows immediately from the definition, okay? It's much stronger than the definition. So you enabled of X, that's U1, enabled of X. Let U be enabled of X, so here's U. So we're proving this direction, no? So X is the limit point, no? So U intersects A, okay? U intersects A, this is not empty. So let, choose a point which call A1, U1. That's U1, the first one, if you want U, equal U1. So choose a point A1, okay? So A1 is here somewhere, you make that picture, you see in the picture, A1, yes, A1, right? So X is T1, how stuff, but here T1 is sufficient, no? X is T1, what does it mean? X has an A vote, which does not contain A1. Small X has an A vote, U2, which such that, which does not contain such that, what is the first point? A1 is not an element of U2, right? And we can assume U2 is in U1, we can assume. So I prefer to write in this, it seems that U2 is contained in U1. Why can we assume that, U2 in U1? Well, if not, what, sorry? Yes, if not, you take the intersection, okay? The intersection of finally many neighbors is always a neighborhood, so you make it smaller. You just make it smaller, okay? So can apply U2, okay? And now that's honestly proof, right? So we have here the smaller neighborhood. Which is U2, so we make a picture, now it's clear, no? So the proof is clear. So now U2, intersection A is not empty, no? Because it's a limit point, oh, it's a limit point, no? So let A2, okay? So we find, sorry, that one is, so here's A2. Well, now I should make bigger this picture, but A2, okay? So recursively, we construct the sequence, okay? So recursively, define, so you go on with A3, no, A4, A5, define An, sequence. An, all distinct, okay? Because we avoid everything what was before, all distinct. In, where everything is in U intersection A, no? They're always smaller, smaller, smaller, smaller. That's the proof, okay? That's the constructed proof. That's a nice proof, okay? You make a picture, big picture, no? And then you have your neighborhood, so you have the first point, you have the smaller neighborhood, you have the second point, you have the third neighborhood, you have the third point. And so you see the points, no? What you cannot say is that this converges to X, okay? That's not true in general. Then we need something much stronger, okay? Another condition. We don't talk about conversion. We have a sequence here, and there is here, but convergence is not clear. Why not? It's not clear because there may be too many neighborhoods at this point, okay? There may be neighborhoods which have nothing to do with these neighborhoods which are somewhere in some other, okay, too many neighborhoods. This will be an important condition for the space, okay? When we want all the convergence, okay? That we will see, but here we don't have. But still we have infinitely many points, okay? So it's a little bit more limited. So there's a second proof from the book. So I give two here now, because it's more interesting. So we still, two I. So we still have this direction, okay? We're still in this direction. So this is from the book. They're the same equally easy, but... So let's suppose I'm also here. X, so what is our sequence? It's a limit point, no? We have a limit point of A. So we have a limit point, X, as before. And U is the neighborhood. And suppose U is the neighborhood. And U is the neighborhood of X, such that the intersection is finite, okay? Such that U intersection A is finite. But I don't know if X is in A, okay? I don't want X here. So I take A minus, maybe X is not in A, okay? So A minus X is A, okay? But this is finite. So I can, it's a finite set. X in A, A1, I don't know, how did you call it? X1, Xm. Well, I prefer A. We are in A anyway, no? So A1, Am. It's finite, finite, okay? That's not what we want. However, this is a finite set, no? So this is open. We have T1, T2, house stuff, okay? Here we need T1, okay? So finite sets are open, okay? Because it's T1 or T2, whatever you prefer. T2, house stuff. Final sets are open. So this is, what do we want to prove? I avoided X, right? So this is not X. So this is finite, finite sets are open. So this implies that this is the neighborhood of X. So this is the neighborhood of X. Intersection A minus X is the neighborhood of X. It's open. It doesn't contain X. So it's the neighborhood of X, right? That's our definition of neighborhood, which does not intersect A. It contains X. It's a neighborhood of X. Yes, you're right. Intersection A minus X. So what I want. One second, one second, yeah. I'm tired. Yes, yes, yes, yes. I wrote something wrong.  So this is A1, Am. Not in the best form, okay? Final sets are what? In the house of space? Close. Right? So the complement is open, okay? So this is open. So this is the neighborhood of X. Which does not intersect A. Intersection, which does not intersect A. Because we avoid these points, okay? Intersection of U with A, it might be X. We don't care about X. But the other points are these points. And these we don't have here, okay? We avoid these, okay? So this is the neighborhood of X.  So this means X is not a limit point, right? So this implies X is not a limit point of, of what? Of A, okay?  Well, I prefer the first proof. It's constructive, okay? I made a mistake. I'm going to have to concentrate a little bit now. We need, the points are closed.  The first is constructive. This is sort of, well, you use something and you write something. One second, I don't like how it is written.   I don't know how it is written in the book. But we suppose X is the limit point, then we prove something, and then we write X is not a limit point, okay?  That's not, so I correct this, okay? To write it in a, this is correct, okay? But it's not written in a nice way.  There is a neighborhood U of X, such that U intersects A only in finitely many points, okay? If there is such a neighborhood, then X is not a limit point. So if X is a limit point, then it intersects in infinitely many points. This is better way to write, no?  You write something which you want to prove, then you independently of this, you prove something, and in the end you write the contrary of that, what you wrote before, no? You can, so this is a good way to write, okay? It's clear, no? It's not a contradiction, no? We prove in some sense A plus B, not B, is equivalent to not B, not A, so that's what we prove, okay? It's not really a contradiction. Okay, then what, that's exactly the, yes? You have to prove that every neighborhood has infinitely many points, infinitely many points. So infinitely many points? Yes. That you have to prove, that you still get infinitely many points. That you have to prove, you exclude X, okay? And then you still have to prove you have infinitely many points in the intersection. That's exactly this proof. Okay. It's not, I mean, it's easy, but you have to write something. It's exactly this proof, okay? Okay. Because you have only finitely many, okay? And then X is not a limit point. Here you have to concentrate, okay? So I made some mistakes. The first one is sort of immediate, right? You see these infinitely many points, no? You see how they, you can construct them in some sense, okay? So that's why I give two proofs, okay? The first one is in some sense nicer. This one is more sophisticated, no? You use points are closed. You're not, you're, okay? You have to write something, okay? Okay? Yes. But still the interesting point is we have a sequence now, infinitely many points now, but we don't have convergence in general, okay? We will see examples. And that's interesting point. Still we cannot suppose that we get a sequence in A which converges to X, okay? That's what we want to have maybe, but that's not true. So we will see examples that this is not true. You don't get a sequence which converges. In A which converges to X. That, there you need some other condition, okay? Some stronger condition. Sorry? Yes. Yeah. In the closure, a limit point, let's say. Yes, in the closure or limit point. And if it's in A itself, you have a sequence now. Constant sequence. A is almost the same point. So interesting point if it's a limit point, okay? And then you want a sequence in A which converges to this point, okay? Maybe you want, but you don't get that, okay? No, one second. If it's an isolated point, it's a limit point only if it's in A. No, no, it's not in A point. If this is A, then this is not a limit point. It's not a limit point. So you have this. We are now two here, okay? Schematic picture. So you have here neighborhood which intersects. So A is this. A is this and this point. Isolated points are not limit points. Yeah. If it's in A, it's in the closure, of course. But it's not a limit point. Isolated points are not limit points, okay? So it's not isolated. But still, we will see that you don't get the sequence. Okay, so now let me go on because... So this is an interesting condition, hostoff. Now continuous function. We have to talk, of course, about continuous functions. Well, continuous functions. Yeah, functions, maps, whatever. So let me give the definition. The definition is very easy. So, definition. X and Y are topological spaces. A function, a map, F from X to Y is continuous. That's the definition. If pre-images of open sets are open. Pre-images. If pre-images of open sets. In other words. So this is a fast way to say that, okay? If U, if V is open in Y, then... What is a pre-image? F minus one, no? Then F minus one of V. So F minus one of V. So this is a pre-image. It means all points. So the definition, of course. F minus one is a double, no? This is not the inverse function. It might be also the inverse function. It's bijective, no? And then it's the same, okay? But in general, it's only the pre-image. It's not the inverse function. Pre-image. So these are all points X in X. Such that F of X is in V. That's the pre-image of V, no? That's the definition. So if V is open, then F minus Y is open. And pre-images of open sets are open. Well, we will say that anyway, but not now. This is the definition. The definition is also open sets, okay? No, for open for some reason. Yeah, close. You suspect maybe it's close, no? It's okay. You get some compliments, no? And you look a little bit around. And then it's true for close. But the definition is open, no? The proof that it's equivalent is easy, okay? You go to compliments, okay? Open and close are compliments. And so the pre-images of complimentary sets are complimentary or something like that, no? And then it seems to be, okay, yes. So this I like to distinguish, call it the global definition of continuity, okay? Global. You have no point here. You have, for each open set in Y, pre-images open. So this is a global definition. Because then we have a local definition, no? This is a global definition, okay? So let me call this I, okay? And now I have a second, two I. So local definition. Local means in a point, okay? So f is continuous in a point, x in x, small x in capital X. f is continuous in a point, x in x. So this is epsilon delta, except we don't have metric spaces, no? So if for every neighborhood v of f of x, if for every neighborhood v of f of x, there exists a neighborhood u of x, such that, such that, such that. f of u is continuous in v. So in analysis, in real analysis, this is given epsilon neighborhood, epsilon ball of f of x. You find a delta neighborhood, a delta ball in x such that this holds, okay? But we don't have epsilon delta here, because we have general topological space. So this is a local definition of continuity, no? In x, local definition. So this is a local definition of continuity. So this is continuous in a point. And this is just continuous, no? This is very short, no? Prematures of open sets are open. What is the relation between these two? Well, that's what you suspect, no? So proposition. f is continuous. f from x to y is the same notation, almost is continuous, if and only if. f is continuous in every point x and x. This is a little bit more general than this local, no? You can say in some point it's continuous, in some point it's not continuous, so you have the points where it is continuous, no? So it's a little bit more general, no? So f is continuous, f is continuous in every point, x and x. So that's the proof. I don't know what is the, I mean, now what in general you take as a condition, as a definition in analysis, epsilon delta, no? But also this one. Prematures of open sets are open, so both. I'm not epsilon delta. So we have two directions. This is, f is continuous. So we have to prove this one, okay? This, we have to prove this. So v is a neighborhood of f of x. So let x be an x. And v a neighborhood of f of x. Then what is u? What you take for u? This is trivial, this is real. What is u? So u, so this is a definition, no? It takes a preimage, f minus one, no? So this implies f minus one of v. So this is open, no? Enabled is open first. So the preimage is open in x. It takes a whole preimage. And of course f of u is f, f minus one of v, is contained in, not equal to v, contained in v, no? It might not be surjective for something, okay? But it's contained in v. So this is trivial, this direction, okay? A little bit more interesting is the other direction. So which is this one? This one, this one. So now if f is continuous in every point, so f is continuous in every point, yes? So let v be open in, so I use almost this notation again, open in, but we have no pointer. Let v be open in y. You have to prove that the preimage is open, okay? The preimage is open in x. No, that's continuity. To take a point x in f minus one of v. So let v be open and take any point in the preimage f minus one of v. This means, of course, f of x. Then f of x is in v. So v is the neighborhood of x, f of x, no. So v is the neighborhood. It's an open set containing f of x. It's the neighborhood of f of x. But f is continuous in this point, okay? f is continuous in x, no? In each point. f continues in x. In x, in this point, x in x. If you write, you see, this is small, this is capital, okay? Otherwise sometimes it's clear from there. f is continuous in x. So this means for the x this, there is a neighborhood u. There's a neighborhood u of x. ux. It depends on x, no? For each x. So we call ux of x such that f of ux is contained in. So what is then f minus one of v? Then the preimage f minus one of v. This should be open, no? And what is it? Just write it this way. You take union over all these, no? Each of these ux, f of ux is contained in v. So each of these is contained in. ux is contained in f minus one of v, no? Okay? Applying f minus one. ux is contained in f minus one of v, no? Right? Because f of ux is in v. So you have this inclusion. And the other inclusion, because you have all x in x here, okay? If you have some x in f minus one of v, then you have ux, okay? Which is a label of x. So this is by construction trivial also, okay? Both are trivial. So this means this is union of open set. So open is open in x. And this is the end of this proof, okay? You just formally check both. Yes, thank you. What is u? f minus one of v? u, not x. Yeah, right. Sorry, there's always some small one that's to concentrate on the notation. No, sorry. Get distracted. So x in u, okay? x in f minus, which is f minus one of v, no? It's this one. It is this one. Example. So in analysis, you have, in real analysis, you have continuous functions, differential functions, continuous, no? So you have some nice examples. So I take r. So this is a standard. And I take rl. This is a lower limit. Oh, let's, is this continuous? Ah, identity. Sorry. Yeah, I wrote id, identity. I abbreviate. I forgot to say that. So this is id, see, identity, yes, sorry. Identity map. Yeah, cool. That's, I always abbreviate identity. It's not continuous, no? Not continuous. Yeah, right. I should have started with the other way. From rl to r is continuous, no? Right? So the general principle here, not principle, the general thing is that the identity map from x, t1. No, I write the topology. This is the identity, okay? Sorry. Id is the identity map. From x with one topology to x, the same x, because I have the identity map, no? I need the same x. t2 is continuous, if and only if. Sorry, Mark. So what should I write? t1 is finer than t2, yeah? And here you have, that's rl. It's strictly finer than r, okay? So it's not finer than r. It's strictly finer than r, and r is not finer than rl, okay? It's not finer, yeah. So definition, homomorphism. Here we have one of the important problems of topology, no? Okay? So, definition. f from x to y. So these are topological spaces, okay? It's a homomorphism if f is continuous. In topology we have continuous maps, okay, in general. Homomorphism means isomorphism, okay? For groups, isomorphism, in topology, homomorphism, okay? So bijective, bijective. And that's not sufficient, okay? So what should I add? Yes, and the inverse map. So now it's the inverse map, not the pre-image. The inverse map, f minus one. This is also f minus one. From y to x is also continuous. And this is not automatic, okay? Why not automatic? Because we have an example before, no? The identity is from rl to r is continuous, bijective, but the inverse map, which is identity from rl to r, is not continuous, okay? So it's not automatic. We have to write it down here, okay? And for homomorphism we want that. And this is an interesting problem in topology. So this is a homomorphism problem, no? That's a homomorphism problem. This you can say is one of the central problems in topology, of topology, in topology. The homomorphism problem says that given two topological spaces, x and y, decide if they're homomorphic or not. Very natural, no? Given two topological spaces, x and y decide whether, if, whether, x and y are, sorry, if there's a homomorphism, then you say x and y are homomorphic, okay, of course. If there is a homomorphism, then you say x and y are homomorphic, okay? I don't write it, but it's clear, okay? Homomorphic groups, homomorphic spaces, okay? So given two topological spaces, it decides whether x and y are homomorphic. So I can write it here, are homomorphic. Homomorph, homomorphic. That means there exists a homomorphism, no? There exists. That's the central problem of topology. This one, okay, you can say. There exists a homomorphism. There are f from x to y. There are two aspects of this, okay? Two aspects. If you're given two, for example, examples. Well, I write minus one, one. x is minus one, one. The real interval, okay? In R, right? This is the real interval, okay? Standard topology, no? Standard topology, from analysis. And R, the reals itself. So these are two spaces and you say, are they homomorphic or not? What would you say? No? Not the same, they are homomorphic. Let's say, sorry. They are homomorphic. And how do you prove that? Construct the homomorphism. That's analysis, okay? The what? Yes, yes. It goes minus one, minus pi over two, or pi over two. All intervals are homomorphic, okay? Yeah, the tangent is okay, yes. So, construct the homomorphism. There are many ways to construct the homomorphism, no? Construct the homomorphism. That's the only way to prove that. The tangent is okay, no? But it's not minus one, one, it's minus pi over two, pi over two, okay? But any two intervals are homomorphic, okay? Homomorphism, by the way, is an equivalence relation, no? Homomorphism, so a remark, maybe trivial, but homomorphism is an equivalence relation and Buster. It's an equivalence relation. On what? On topological spaces, on all, okay? It's an equivalence relation. So intervals, all intervals, okay? So, and here, you can, as you say, I can take this interval, no? They are homomorphic, okay? All intervals, all open intervals are homomorphic. All open intervals are homomorphic. You can get some linear homomorphism, okay? First, you can center around zero and then you have a linear, okay, multiplication with some constant, no? Okay, we have two intervals. First, you translate to zero, right? You have an interval somewhere out in R, in the reals, no? Then you have the midpoint, okay? And then you get the translation, so the midpoint is zero, okay? That's what you do with both intervals and then you have two intervals around zero and then you just multiply with lambda, the smaller, where lambda is the ratio of... Yeah, that's analysis. It's not so much our, okay? But the idea sort of... So all intervals are homomorphic. And now comes the intriguing question. Better like, so problem, examples. Examples, that's the first example, okay? And now two are R, again, the standard. And now I write RL, yeah. RL is a strange space, no? So you say, R is a standard, R, RL, that's some strange topology. So you say not homomorphic. But then you have a problem, no? Now you have a problem, yeah? It's easy, no? Not easy, but you have two spaces. If they are homomorphic, you think and think and then you see the homomorphism. Here, you have to exclude the homomorphism, okay? That's a different order of difficulty, no? You have to exclude. So in general, how do you exclude that? The only way, you need some property, some topological property, okay? Which one has the other not, okay? So need some... One needs... That's the point. Some topological property. So this I don't define, but we give examples, okay? Topological property, okay? Which one has and the other not. A topological property is... Which one of these has and the other does not have, okay? And the other not. So we don't have many topological properties, okay? Sorry? One point in RL is not... Oh, that's not true. It has not a discrete topology, RL. Be careful. Points are not open in RL. That would mean the discrete topology. Okay? Points are not open. RL is not the discrete topology, okay? It's not... So you may say I met at RD, the discrete one here, okay? So let me write topological property. So a certain example, then you say R and I take RD, discrete, okay? And then you would say they are not on your morphic, not on your morphic. So how do you prove that? You say it's easier this one now. Why... How do you prove that R... R is not discrete, okay? It's not the discrete top... R has not the discrete topology. Now that's the point. You say points are not open, okay? In R. So this is easy, no? Since... That's what you say, okay? Since points are not open in R. Are not closed in R. Not open in R. Open, open, open. Since points... If there would be a homeomorphism, no? Then points are open in RD, discrete. And so the image of a point is a point and a homeomorphism makes open sets to open sets, okay? But this is... So this is what you say, okay? Since points are not open in R. R has not the discrete topology. Now that we know, okay? R has not the discrete. So that's the property, topological property. Points are open. That's the discrete topology. That characterizes the discrete topology, no? But that's the topological property. Since R has not the discrete topology. That's clear. Being a discrete space. Exactly, yes, right. That's the topological property. Being discrete, okay? Yeah, this is different. Because we don't have many properties so far, no? And yes? Yeah, that... So in R, L, certainly we have open sets which have a minimum, okay? But this is a notion of order. This minimum, no? And now you say this open set has a minimum in R, L. And now you apply F. And then you have an open set in R. The standard rules, right? And then they say this has not a minimum, okay? But why should a minimum be preserved by such a map? It's not order preserved or whatever. This set, 0, 1, they are open in R, L, no? Of course this is not open in R, that's clear. But the image in R, you don't know the homomorphism. The map might be very complicated. It might not preserve the order. So then you have to prove that if this has a minimum then the image should have also a minimum in R. But then I'm not convinced that you can... There's an easy proof to do that. Here, this 0. No. No, no, the map is very complicated, okay? It's the identity map. The identity map is of course not. But this map is extremely complicated. It's on rational numbers, irrational. It's a big distortion map, okay? Which makes everything complicated, okay? But it's subjective. Well, you can try to... That's an idea of course, okay? You say this is a property... Well, a property of... The point is that this notion of minimum, okay? Or maximum. It's not purely topological, no? It is order, okay? It's not purely topological. It has something to do with order, so we have more structure, okay? That's the problem, right? It's not purely topological, okay? So there's a problem. So the topological property is house stuff, no? But R is house stuff. RL is house stuff. That's the only we know. And so now I give you the property, okay? It's a good point to introduce the property, okay? To prove that. Well, what short? You have to use some property. Which one? Ah, you're talking about... Yeah, that's a good point probably. You will be talking about connectivity, okay? We will forget about connectivity at this point. Yeah, but that's... You're talking about R is... Yeah, that's true, but that you have to prove. I mean, this is connectivity. And this is... That we will do anyway, but you... That's true. You're right. So connectivity distinguishes, okay? Yes, yeah, connectivity. But then you have to prove that R is connected. Okay. Now that means R is... That's the definition of connectivity. The only empty... The only open and closed sets are... That you have to prove. If you prove that, then you are done. But that you have to prove. That means connected. That's the definition of connected. And then you have to prove R is connected. That's true, of course, okay? That's the topological analysis. And we will do connectivity, of course. Now that's the topological property. Connectivity distinguishes, okay? But then you have to prove the theorem first. R is connected. The theorem. Which is not completely trivial, okay? This is on the basis of everything in analysis, no? This is... Of course, it's... Yeah, that's a good point. The connectivity we will do later, okay? Not today, so... So we need something easier. So it's a good point to do this, because we're talking about bases. Okay? So we have bases. We know what is the basis for topology. And so the fact is, R... So has a countable basis. So this is the definition of first countable. No, no, of second countable. So R has a countable... What is the countable basis of the wheels? So what is the basis of the wheels? The bases are all open intervals. And what is the countable basis? With... Rational points, yeah? So a countable basis. All intervals... B. No, no. All intervals with rational inputs. B is equal... to... AB. A smaller B. AB and Q. This is countable. This is a countable basis. Well, you make the usual picture now. This is exercise. This is also one of the exercises. It's a book, by the way. So you have to... You have to prove this is the basis. How do you do this? You have the standard basis, all open intervals. And you have to compare. And then you apply the lemma. So you take... This is an arbitrary open interval. Okay? So what is it? CD. This is an arbitrary open interval. This is an exercise. And what... So we have... Now we have this collection here. C, okay? The lemma says... This is a basis for our topology. If, given any... open set, basis element is sufficient, given any point here, we find some of these which contains the point and this contains here. Then this is the basis for our topology. That's the lemma, okay? And so how do we do this? Yes, the density. So this interval contains... Any interval contains point of Q is dense. Q is dense. So this contains... So we have this interval, no? No, sorry, which one? And they contain rationals. Any interval contains rationals. And so we have AB, no? And rationals, and so we have X. So if you have to write this proof, then... we have this, okay? And apply a lemma, okay? So the lemma says this is the basis. Okay? It's a countable, it's clear, no? It's countable. Q is countable. So this is like... which is countable, no? Q times Q is countable. Countable, yeah. Countable set times countable set is countable, no? Well... That needs some proof also, but... RL does not have a countable basis. If this is the case, no? Then we are done. Then they are different, okay? So this implies R is not... So this implies R... So homeomorphism, you write it like this, no? Homeomorphism. But it's not homeomorphism. It's not homeomorphic to RL. Because this has a countable basis. This does not have a countable basis. But if you have homeomorphism, they are the same, no? Homeomorphism means we have... the properties are exactly the same, no? This is a bijection, and it's just other names for the points, okay? But these are the same space, the names for points, okay? That's the homeomorphism. So RL is not a countable basis. And that's a nice proof. So proof. That's in the later chapter of the book. But it has only to do with basis, so it's a good point here, okay? We define basis, so proof. It's in chapter... We are doing chapter two, okay? This is chapter four. So it gives a definition in the second of the property. So let's be... be any basis... We don't know anything. Any basis for RL. So B is not this basis, you know? Any basis. And what we have to prove is that B is not countable, okay? No countable basis. But we don't know what is this basis, no? What? So the proof is easy, but nice. So we have X... X plus one. This is open in RL, no? That's a standard basis. Not our basis, B, you know? By the way, the standard basis... This is a standard basis. This is not countable, no? Because we have too many points here, no? We have all reals here. We cannot, okay? So it's not so clear. You can try to do rationals here, now, okay? That is not the right topology. Okay? You need all reals here. So this basis you cannot restrict to a countable, okay? That doesn't work. Well, anyway, this is open in RL. And X is here, of course, no? So we have a basis now, B. This we don't know, okay? But anyway, we have an open set, we have a point. What means basis? We find the basis element which contains the point is contained in the open set, no? That's a usual stuff. So there is, there exists BX in B. So it depends on X. Such that X is in BX, which we don't know, but it exists. And this is contained in X, X plus one. This is an open set, no? Okay? So we get, for each X, we can do this. We can find this, okay? So we have, for each X choose BX for each X, okay? Choose some BX for each X. Choose a BX for each X in X. X in X is where? In RL, no? It's a set there. So such and BX for each X in RL, okay? So we have, and then we have a function from R to B X, and for each you have such BX, okay? For each X you choose this, because it exists. And now we want to prove that this function is injective, okay? That they are all different to BX. If this is the case, then R is, so we have phi, function phi. This is not countable, no? If this is injective, then this is not countable. Right? So let's see. Suppose X and Y are in R and X is different from Y and X is smaller than Y. Doesn't matter, Y is smaller than X. X smaller than Y. One is smaller, okay? They are different anyway, okay? And then we want to compare BX and BY. No? So BX and BY, right? For each X we have BX. For Y we have BY, okay? So we compare BX and BY. So BX and BY. And we want to compare this. We want to prove they are not equal, no? That's injectivity, okay? That's injectivity. They should not be equal. So BX is contained in X, X plus 1. BY is contained in Y, Y plus 1. Right? Now X is smaller than Y. X certainly is contained here. But X is smaller than Y. X is smaller than Y. So it's not here. So X is here but not here. And so this implies BX is different from BY. And this implies phi is injective. So then phi is injective. And R is uncountable, no? And that means, so R is uncountable. And this implies that B is uncountable. Not countable. So this basis was not countable. So this implies there is no countable basis. Okay? Because this is arbitrary basis, no? So RL has no countable basis. If you want to read this, this is chapter 4 of the book, okay? This proof. And the definition, so I give now the definition. So this is proof, okay? R, the standard, has a countable basis. RL intervals. RL, the strange topology, lower limit has no countable base. That's not every countable base, okay? So they are different spaces. Maybe that's another proof is of course using connectivity, okay? That we will prove later. So there are various properties, no? But we have to find one. So definition. This is an interesting property. So definition. The topological space B, X is second countable if X has a countable basis. It's a topology of X, of course. But if X has a countable basis, that's the definition. Well, this is a strange definition. I mean, we may just say countable basis, no? But we can take second countable. The point is, the second countable you suspect that there's first countable. So you have another property. There will be another property, okay? That's why it's second countable, no? If there's no first countable, then you just say countable basis. That's it, no? You say sometimes second countable because there's also first countable. It is another important condition for a topological base. But we will see much later, okay? So this is second countable. And this is a nice proof. RL is not so corollary. It's the same as before. RL is not second countable. That's the way to say it. RL is not second countable. The interesting point is that we talk what is a topological property, no? What are nice topological properties, okay? This is the main point of general topology. The topological properties, okay? What are the important or interesting topological properties of topological spaces, okay? Whereas the main point of algebraic topology is not topological properties, but topological invariance. Numbers, groups, vector spaces. For example, an algebraic invariant. Let's talk about one minute, it's finished now. One minute about groups, okay? There's a big difference here. If you have a homomorphism of groups, you will do algebra, but you know what the homomorphism of groups is, okay? For homomorphism of groups, this is bijective, okay? Then you have the inverse map. And this is a homomorphism, okay? Automatic. If you have a linear map between vector spaces, and this is bijective, then the inverse map is linear. That's automatic. You don't have to, you can easily prove that, okay? In algebra, if you have bijective and preserve the then the inverse map is okay, it's the same, okay? Isomorphism, it's a quote. Isomorphism, no? And if you want to distinguish two groups. So here you have the homomorphism problem for topological spaces, okay? Decide if two topological spaces are homomorphic or not, okay? If you want to prove that they are not homomorphic, you need something. A topological property or a topological invariant. If you want to prove that two groups are not isomorphic, you need something. You need, for groups, it's more algebra. So you need maybe a group, not topological group invariant, okay? Invariant of groups. What is the easiest invariant of a group? Yeah, for example, yes. That's... Yeah, exactly. Invariant is the order. How many elements, okay? If you have a group with 10 elements and you have a group with 20 elements, they are not isomorphic, clearly, no? Algebraic invariant, okay? The order of a group, this is more already a little bit more sophisticated, no? You have elements of order two in one, no elements of order two in the other, so they cannot be isomorphic, okay? Because isomorphism preserves order, no? Yeah. And so general topology is largely the study of topological properties, okay? Topological properties. House of first count of the, second count of the, connected, compact, the various notions of this, okay? Not just one. An algebraic topology is studying topology invariants, okay? So this will be groups. We have studied in the second part, fundamental group, okay? And then there are homology groups, co-homology groups, homotopy groups. Yeah, so there are many algebraic in there, okay? So this is one difference between these two. General topology and algebraic. That's algebraic topology, algebraic invariants, okay? Dimension, maybe. But dimension is also a number, no? So dimension, as if a group is the order, no? It's a number, just a number. The dimension, but dimension is difficult to define. For vector spaces, the dimension is the first invariant, okay? For vector spaces, the first invariant is the dimension, no? Yeah, here goes both directions, no? I'm only wondering, if they are isomorphic, they have the same dimension. And then it happens for vector space. If they have the same dimension, then they are also isomorphic. So this is a complete invariant. But that's... difficult to find. Vector spaces are easy in some sense, no? Groups are difficult, okay? There are many... You don't have anything equivalent for groups or for topological spaces, okay? So in this sense, vector spaces are easy, yes. You have to write it, okay? And comment. So I made a small list. You have to write something, okay? You have to start writing, okay, in English. And next week, I... collect... exercises. Because the final exam will be exercises like this, okay? And then you have to write in English, okay? You have to be careful and... So I give a couple of exercises. First group. These are... I comment on the exercise. What is easy, what is not easy, okay? For exercises. So page 83. And here, 6. 6. This I introduced because there is... introduced, which we can do here, but you have to... there's a new topology on the reals, okay? Which is also some nice property. So, and this is 7. So 7... only for T1 and T2. T1 is R. And T2 is... So this is the other topology, RK. It's called RK, K-topology. You have to read what is the... this topology on R, okay? And then you have to compare these two. And this is also for RK. This is also an exercise for this topology, okay? So I give this exercise so you... And here, you just give the answer, okay? Because you don't know how to write, maybe, okay? You have to compare topologies. And then we make my picture, okay? And you say what happens, okay? You just think and if you understand, then it's okay, right? Sorry? Yes, in the book, of course. So just give the answer. Because you don't know how to write that in a good way, maybe, okay? So they say this topology is not comparable or this is finer or strictly finer, okay? What happens? You want to know what happens. And here it's similar, okay? You compare RK with R, I think, okay? So you just think what happens, okay? Which is finer? Are they comparable? No? It's an exercise to think not so much to write, okay? Then there are some formal exercises and here you have to write. Give the answer for this. Then page 92 Exercise 4 Here you write. These are easy exercises. You should write, okay? In a good way. And also page 101. Page 101. So these are very easy exercises. 10. Except you have to think how to write that in a good way. That's the main point. 10, 11, 12, 13. So these exercises you write. Write the proofs, okay? Here you have to write the proofs. Write down the proofs. And they are easy. They are very easy. So it should be no problem. Except for the English, maybe, okay? And then there's the last one which is like this one where you have to think a little bit. And this is strange. Page 101 And this is exercise number 18. I hope I don't mix up. First of all, can you show me what you did? Yeah, 18 is a long exercise. Determine the closure of the order square. This is an exercise for this is an exercise for for I square orders, which is important, okay? That is just give the answer. Because it's too long to write this, okay? You have to think what happens, okay? And you just give the answer. Like this one, okay? These are the formal exercises. Just give the answer. That means you have to understand what happens, okay? You think about it and just give the answer. Because otherwise it's too long. There are five cases, no? It's interesting because you have to work with this space, okay? The order square and here you have to work with this space. Okay? So these two exercises are because you have to know what this is and what this is, okay? So I give these two exercises. The other ones are easy. Very easy. Okay, short. But you have to write in a nice way. That's the problem, no? You have to concentrate on that. Okay? So these are the exercises for next week. And you write down. You have to write down because sorry? Tuesday. Yes, yes, exactly. One week. But you have to start writing. That's very important, okay? Because the final exam is eight exercises, okay? Eight, nine exercises. If you never, if you don't do that regularly, you have a problem, no? In some sense, analysis and algebra are more computational, okay? It's not computational. You have to understand, okay? What happens? So this is it's a little bit different. Okay? The approach. The analysis you have of proofs also, until algebra, of course, but you have also computations, no? Here in topology, it's more logic or proofs, okay? Mainly proofs and constructions, okay? And then you have to write in a good way, okay? So you have to concentrate on the way how to present.


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