Chapter 18. The Woman Who Did Not Dare Polly wrote enthusiastically. Ned answered satisfactorily, and after much corresponding, talking and planning, it was decided that Tom should go west. Never mind what the business was, it suffices to say that it was a good beginning for a young man like Tom, who, having been born and bred in the most conservative class of the most conceited city in New England, needed just the healthy, hearty social influences of the west to widen his views and make a man of him. Of course, there was much lamentation among the women, but everyone felt it was the best thing for him. So while they sighed, they sued, packed visions of a brilliant future away with his new pocket-handkerchief, and rejoiced that the way was open before him, even in the act of beduring his boots with tears. Sydney stood by him to the last, like a man and a brother, which expression of Tom's gave any infinite satisfaction, but Will felt entirely consoled for Ned's disappointment at his refusal to go and join him, since Tom was to take the place Ned had kept for him. Fortunately, everyone was so busy with the necessary preparations that there was no time for Romans of any sort, and the four young people worked together as soberly and sensibly as if all sorts of emotions were not bottled up in their respective hearts. But in spite of the silence, the work, and the hurry, I think they came to know one another better in their busy little space of time than in all the years that had come before, for the best and bravest in each was up and stirring, and the small house was as full of the magnetism of love and friendship, self-sacrifice, and enthusiasm as the world outside was full of spring sunshine and enchantment. Pity that the end should come so soon, but the hour did its work and went its way, leaving a clearer atmosphere behind, but the young folks did not see it then, for their eyes were dim because of the partings that must be. Tom was off to the west, Polly went home for the summer, Maud was taken to the seaside with Belle, and Fanny left alone to wrestle with housekeeping, help, and heartache. If it had not been for two things, I fear she never would have stood a summer in town, but Sidney often called, till his vacation came, and the voluminous correspondence with Polly beguiled the long days. Tom wrote once a week to his mother, but the letters were short and not very satisfactory, for men never do tell the interesting little things that women best like to hear. Fanny forwarded her bits of news to Polly, Polly sent back all the extracts from Nett's letters concerning Tom, and by putting the two reports together they gained the comfortable assurance that Tom was well, in good spirits, hard at work, and intent on coming out strong in spite of all obstacles. Polly had a quiet summer at home, resting and getting ready in mind and body for another winter's work. From the autumn she tried her plan again, to the satisfaction of her pupils, and the great joy of her friends. She never said much of herself in her letters, and Fanny's first exclamation when they met again was anxious.  Now I'm only tired, had a good deal to do lately, and the dull weather makes me just a travel-brew. I shall soon brighten up when I get to my work again, answered Polly, bustling about to put away her things. You don't look a bit natural.  Persisted Fanny, troubled by the change, yet finding it hard to savour in a distant lay. Polly did not look sick, though her cheeks were thinner and her collar paler than formerly, but she seemed spiritless, and there was a tired look in her eyes that went to Fanny's heart. I'm all right enough. As you see when I'm in order, I'm proper glad to find you looking so well and happy. Does all go smoothly, Fanny? asked Polly, beginning to brush her hair industriously. Answer me one question first, said Fanny, looking as if a sudden fear had come over her. Tell me, Julie, have you never repented of your hint to Sidney? Never, quite Polly, throwing back the brown veil behind which she had half-hidden her face at first. On your honour, as an honest girl. On my honour, as anything you please, why do you suspect me of it, demanded Polly, almost angrily? Because something is wrong with you. It's no use to deny it, for you've got the look I used to see in that very glass on my own face, and I thought he cared for you. Forgive me, Polly, but I can't help saying it, for it is there, and I want to be as true to you as you were to me if I can. Fanny's face was full of agitation, and she spoke fast and frankly, for she was trying to be generous and found it very hard. Polly understood now and put her fear at rest by saying almost passionately, I'll tell you, I don't love him. If he was the only man in the world, I wouldn't marry him, because I don't want to. The last three words were added in a different tone, for Polly had checked herself there, with a half-fried look, and turned her way to hide her face behind her hair again. Then, if it's not him, it's someone else. You've got a secret, Polly, and I should think you might tell it, as you know mine, said Fanny, unable to rest till everything was told, for Polly's men had troubled her. There was no answer to her question, but she was satisfied in putting her arm around her friend, she said in her most persuasive tone, My precious Polly, do I know him? You have seen him. And is he very wise, good, and splendid, dear? No. He ought to be if you love him. I hope he isn't a bad, quite Fanny, anxiously still holding Polly, who kept her head obstinately turned. I'm suited, that's enough. Oh, please, just tell me one thing more. Don't he love back again? No. No, don't say another word. I can't bear it. And Polly threw herself away as she spoke in a desperate sort of tone. I won't, but now I'm not afraid to tell you that I think. I hope I do believe that Sidney cares a little for me. He's been very kind to us all, and lately he has seemed to like to see me always when he comes, and miss me if I'm gone. I do not dare to hope anything till Pavel observed something in his manner and teased me about it. I tried not to deceive myself, but it does seem as if there was a chance of happiness for me. Thank heaven for that! Quite Polly with the hardiest satisfaction in her voice. Now come and tell me all about it, she added, sitting down on the couch with the air of one who has escaped a great peril. I've got some notes and things I want to ask your opinion about. If they really mean anything, you know, said Fanny, getting out a bundle of papers from the inmost recesses of her desk. There's a photograph of Tom, came in his last letter. Good, isn't it? He looks older, but that's the beard in the rough coat, I suppose. Dear old fellow, he is doing so well I really begin to feel quite proud of him. Fann tossed her the photograph and went on remorging for a certain note. She did not see Polly catch up the picture and look at it with hungry eyes, but she did hear something in the low tone in which Polly said, I don't do him justice. And glancing over her shoulder, Fann's quick eye caught a glimpse of the truth, though Polly was half turned away from her. Without stopping to think, Fann dropped her letters, took Polly by the shoulders, and cried in a tone full of astonishment. Polly, is it Tom? Poor Polly was so taken by surprise that she had not a word to say.  Oh Polly, I am so glad. I never thought of it, you are so good.  Couldn't help it, try it not to, but it was so hard, you know, then, you know, so distraught voice from the depths of the very fuzzy cushion which Tom had once condemned. The last words and the appealing hand outstretched to her, told Fanny the secret of her friend's tender sympathy for her own love troubles, and seemed so pathetic that she took Polly in her arms and cried over her, in the fond foolish way girls have of doing, and their hearts are full, and tears can say more than tongues. The silence never lasts long, however, for the feminine desire to talk it over usually gets the better of the deepest emotion. So presently the girls were hearted it, Polly very humble and downcast, Fanny excited and overflowing with curiosity and delight. Really, my sister, your dear thing, how heavenly that will be, she cried, it never will be, answered Polly in a tone of calm despair. What have they prevented? Maria Bailey must have tried to reply. What do you mean? Is she the rustling girl?  Too late, let me tell you, is that door shut and more safe? Fanny recanointed and returning listened breathlessly, while Polly poured into her ear the bitter secret which was praying on her soul. Hasn't he mentioned Maria in his letters? Once or twice, but sort of chokingly, and I thought it was only some little flotation. He can't have time for much of that fun, he's so busy. Ned writes good gossipy letters, I taught him how, and he tells me all that's going on. And he'd spoken of this girl several times, they bought with her mother, you know, I asked about her quite carelessly, and he told me she was pretty good and well educated, and he thought Tom was rather smitten. That was a blow for you see, Fen, since tricks broke the engagement, and it was not wrong to think of Tom. I let myself hope, just a little, and was so happy. Now I must give it up, and now I see how much I hoped, and what a dreadful loss it's going to be. Two great tears rolled down Polly's cheeks, and Fanny wiped them away, feeling an intense desire to go west by the next train, with a Maria Bailey with a single look, and bring Tom back as a gift to Polly.  I know you were very good to him. You are to everyone, my precious, and I knew that he was fond of you as he is of me, fond of anything, because he thinks you are perfect, but still I never dreamed of his loving you as more than a dear friend. He doesn't, said Polly. Well, he ought, and if I could get hold of him, he should. Polly clutched Fen at that, and held her tight, saying sternly, if you ever breathe a word, drop a hint, look and look that will tell him or anyone else about me. Yes, as sure as my name is Mary Milton, I proclaim from the housetops that you, like us, Polly, got no further, for Fen's hand was on her mouth, and Fen's alarmed voice vehemently protested. I won't. I promise solemnly I'll never say a word to a mortal creature.   It makes me wild just to think of it. Oh, Fen, I'm getting so ill-tempered and envious and wicked. I don't know what will happen to me. I'm not afraid for you, my dear, and I do believe things will go right because you are so good to everyone. I don't could help adoring you. I don't see. I know you would if he had stayed at home longer after he got rid of tricks. It would be the making of him. But though he is my brother, I don't think he's good enough for you, Polly, and I don't quite see how you can care for him so much when you might have had a person so infinitely superior. I don't want a superior person. He tired me if he was like he is. Besides, I do think Tom is superior to him in many things. Well, you need not stare. I know he is, or will be. He's so different, and very young, and has lots of thoughts, I know, but I like him of the better for it. And he's honest and brave, and has got a big, warm heart. And in what I have him care for me, then the wisest, best, most accomplished man in the world, simply because I love him. If Tom could only have seen Polly's face when she said that. It was so tender, earnest, and defiant that Fanny forgot the defense of her own love and admiration of Polly's loyalty to hers. For this faithful, all-absorbing love was a new revelation to Fanny, who was used to hearing her friends boast of two or three lovers a year, and calculate their respective values, with almost as much coolness as the young men discussed the fortunes of the girls they wished for, but could not afford to marry. She had fought her love for Sydney very romantic, because she did not really care whether he was rich or poor, though she never dared to say so even to Polly for fear of being laughed at. She began to see now what true love was, and to fear that a sentiment which she could not conquer was a treasure to be accepted with reference, and cherished with devotion. I don't know when I began to love Tom, but I found out that I did last winter, and was as much surprised as you are, continued Polly, as if glad to unburden her heart. I didn't approve of him at all. I thought he was extravagant, reckless, and dandified. I was very much disappointed when he chose Triggs, and the more I thought and saw of it, the worse I felt for Tom was too good for her, and I hated to see her do so little for him, and she might have done so much, because he is one of the men who can be led by their affection, and the woman he marries can make or mar him. That's true, great fan, as Polly paused to look at the picture which appeared to regard her with a grave-study look, which seemed rather to belie her assertions. I don't mean that he's weak or bad. If he was, I should hate him, but he doesn't need someone to love him very much, and make him happy, as a good woman best knows how, said Polly, as if answering the mute language of Tom's face. I hope Maria Bailey is only thinksher, she added softly, for I could not bear to have him disappointed again. I dare say he don't care thick for her, and you are only borrowing trouble. But do you say that answered when you asked about this inconvenient girl, said Fanny, turning hopeful all at once? Polly repeated it, and added, I asked him another letter if he did not admire Miss B as much as Tom, and he wrote back that she was a nice girl, but he had no time for nonsense, and I need not get my white kids ready for some use yet, unless to dance at Tom's wedding. Since then he has not mentioned Maria, so I was sure there was something serious going on, and being in Tom's confidence he kept quiet. It does look bad. Suppose I say a word to Tom, just inquire after his heart in a general way, you know, and give him a chance to tell me if there's anything to tell. I'm willing, but you must let me see the letter. I can't trust you not to hint or say too much. You shall. I keep my promise in spite of everything, but it will be hard to see things going wrong when the word would set it right. You know what will happen if you do, and Polly looked so threatening that Fann troubled before her, discovering that the gentlest girls engrossed are more unprocessed than any shrew, for even turtle doves peck gallantly to defend their nests. If it is true about Maria, what shall we do? said Fanny after a pause. Bear it. People always do bear things somehow, and said Polly, looking as if sentence had been passed upon her. But if it isn't, cried Fann, unable to endure the sight. Then I shall wait, and Polly's face changed so beautifully that Fann hugged her on the spot, fearfully wishing that Maria Bailey never had been born. Then the conversation turned to lover number two, and after a long confabulation Polly gave it as her firm belief that A.S. had forgotten M.M., and was rapidly finding consolation in the regard of F.S. With this satisfactory decision the council ended after the ratification of a lawyer league, by which the friends pledged themselves to stand staunchly by one another, for the triers of the coming year. It was a very different winter from the last for both the girls. Fanny applied herself to her duties with redoubled ardour, for A.S. was a domestic man, and admired housewifely accomplishments. If Fanny wanted to show him what she could do toward making a pleasant home, she certainly succeeded better than she suspected. For in spite of many failures and discouragements behind the scenes, the little house became a most attractive place, to Mr. Sidney at least, for he was more the house-friend than ever, and seemed determined to prove that change of fortune made no difference to him. Fanny had been afraid that Polly's return might endanger her hopes, but Sidney met Polly with the old friendliness, and very soon convinced her that the nipping in the bad process had been effectual. For being taken early, the sprouting affection had died easy, and left room for an older friendship to blossom into a happier laugh. Fanny seemed glad of this, and Polly soon set her heart at rest by proving that she had no wish to try her power. She kept much at home when the day's work was done, finding it pleasanter to see dreaming of a book or sewing alone, than to exert herself even to go to the shores. Fan don't need me, and Sidney didn't care whether I come or not, so I'll keep out of the way, she would say, as if to excuse her seeming indolence. Polly was not at all like herself that winter, and those nearest to her saw and wandered at it most. Will got very anxious, she was so quiet, pale and spiritless, and distracted poor Polly by his affectionate stupidity, that she completed his bewilderment by getting cross and scouting him. So he consoled himself with more, who, now being in her teens, assumed dignified heirs, and ordered him about in a style that afforded him continued amusement and employment. Western news continued vague, for fans' general inquiries produced only provokingly unsatisfactory replies from Tom, who sang the praises of the beautiful Miss Bailey, and professed to be consumed by a hopeless passion for somebody in such half-comic, half-tragic terms that the girls could not decide whether it was all that boys mischief or only a cloak to hide the dreadful truth. We'll have it out of him when he comes home in the spring, said Fanny to Polly, as they compared the letters of their brothers, and agreed that men were the most uncommunicative and provoking animals under the sun. For Ned was so absorbed in business, that he ignored the whole Bailey question and left them in utter darkness. Hunger of any sort is a hard thing to bear, especially when the sufferer has a youthful appetite, and Polly was kept on such a short allowance of happiness for six months, that she got quite thin and interesting, and often when she saw how big her eyes were getting and how plainly the veins on her temples showed, indulged a pensive thought that perhaps spring dandelions might blossom over her grave. She had no intention of dying till Tom's visit was over, however, and as the time drew near, she went through such alternations of hope and fear, and lived in such a state of feverish excitement that spirits and color came back, and she saw that the interesting pallor she had counted on would be an entire failure. May came at last, and with it a burst of sunshine, which she had even poor Polly's much enduring heart. Fanny came walking in upon her one day, looking as if she brought tidings of such great joy that she hardly knew how to tell them. Prepare yourself, somebody is engaged, she said in a solemn tone, that made Polly put up her hand as if to ward off in an expected blow. No, don't look like that, my poor dear, it isn't Tom, it's I. Of course, there was a rapture followed by one of the deliciously confidential talks which Boss and Friends enjoy in this burst of tears and kisses, smiles and sighs. Oh, Polly, though I've waited and hoped so long, I could not believe it when it came, and don't deserve it. But I will, for the knowledge that he lost me seems to make everything possible, said Fanny with an expression which made her really beautiful for the first time in her life.  He says that is what made him love me, answered Fanny, never calling her lover by his name, but making the little person a prone on a very sweet word by the tone in which she uttered it. He was disappointed in me last year, he told me, but you said good things about me, and though he didn't care much then, yet when he lost you and came back to me, he found that you were not altogether mistaken, and he has watched me all this winter, learning to respect and love me better every day. Oh, Polly, when he said that, I couldn't bear it, because in spite of all my trying, I'm still so weak and poor and silly.  Thank you all the more, then, for not keeping him yourself, said Fanny, loving the old play of laugh again. That was only a slight aberration of his. He knew better all the time. It was your white cloak in my idiotic behaviour the night we went to the opera that put the idea into his head, said Polly, feeling as if the events of that evening had happened some twenty years ago, and she was a giddy young thing, fond of gay bonnets and girlish pranks. I'm not going to tell Tom a word about it, but keep it for surprise till he comes. He will be here next week, and then we'll have a grand clearing up of mysteries, said Van, evidently feeling that the millennium was at hand. Perhaps, said Polly, as her heart flooded and then sunk, for this was a case where she could do nothing but hope, and keep her hands busy with world's new set of shirts. There's a good deal more of this sort of sudden suffering than the world suspects, for the women who dare, of you, the women who stand and wait, are many. But if work baskets were gifted with powers of speech, they could tell stories more true and tender than any we read. For women often suit the tragedy or comedy of life into their work, as they sit apparently safe in serene at home, yet are thinking deeply living whole heart histories, and praying fiefden prayers while they employ the pretty travels or do the weekly mending. CHAPTER XIX Tom's Success Come, Philander, let us be a-marching, everyone his true love a-searching, would be the most appropriate motto for this chapter, because, intimidated by the threats, denunciations, and complaints showered upon me in consequence of taking the liberty to end a certain story as I liked, I now yield to the amiable desire of giving satisfaction, and, at the risk of outraging all the unities, intend to pair off everybody I can lay my hands on.  That spring the disease broke out with great violence in the Shah circle, causing paternal heads much bewilderment, as one case after another appeared with alarming rapidity. Fanny, as we have seen, was stricken first, and hardly had she been carried safely through the crisis when Tom returned to swell the list of victims. As Fanny was out a good deal with her Arthur, who was sure that exercise was necessary for the convalescent, Holly went every day to see Mrs. Shah, who found herself lonely, though much better than usual, for the engagement had a finer effect upon her constitution than any tonic she ever tried. Some three days after fan's joyful call, Polly was startled on entering the Shah's door by Maad, who came tumbling downstairs, sending an avalanche of words before her. He's come before he said he should to surprise us.  He looks so funny with whiskers, but he's ever so nice, real big and brown, and he swung me right up when he kissed me. Never mind your bonnet, I can't wait.  The sooner it's over, the better for me, was the only thought Polly had time for before she plunged into the room above, propelled by Maad, who cried triumphantly. There he is, ain't he splendid? For a minute everything danced before Polly's eyes, as a hand shook hers warmly and a gruffish voice said heartily, How are you, Polly? Then she slipped into a chair beside Mrs. Shah, hoping that her reply had been all right and proper, for she had not the least idea what she said. Things got steady again directly, and while Maad expatiated on the great surprise, Polly ventured to look at Tom, feeling glad that her back was toward the light and his was not. It was not a large room, and Tom seemed to fill it entirely. Not that he had grown so very much, except broader in the shoulders, but there was a brisk, genial, free and easy air about him, suggestive of a stirring out of door life, with people who kept their eyes wide open and were not very particular what they did with their arms and legs. The rough and ready traveling suit, stout boots, brown face, and manly beard, changed him so much that Polly could find scarcely a trace of elegant Tom Shah in the hearty-looking young man who stood with one foot on a chair while he talked business to his father in a sensible way, which delighted the old gentleman. Polly liked the change immensely, and sat listening to the state of Western trade with as much interest as if it had been the most thrilling romance, for, as he talked, Tom kept looking at her with a nod or a smile so like old times that for a little while she forgot Maria Bailey and was in bliss. By and by Fanny came flying in and gave Tom a greater surprise than his had been. He had not the least suspicion of what had been going on at home, for Fanny had said to herself, with girlish malice, if he don't choose to tell me his secrets I'm not going to tell mine, and had said nothing about Sydney, except an occasional allusion to his being often there and very kind. Therefore, when she announced her engagement, Tom looked so staggered for a minute that fan thought he didn't like it. But after the first surprise passed, he showed such an affectionate satisfaction that she was both touched and flattered. What do you think of this performance? asked Tom, wheeling round to Polly, who still sat by Mrs. Shah in the shadow of the bed curtains. I like it very much, she said in such a hearty tone, that Tom could not doubt the genuineness of her pleasure. Glad of that, hope you'll be as well pleased with another engagement that's coming out before long, and with an odd laugh, Tom carried Sydney off to his den, leaving the girls to telegraph to one another the awful message, it is Maria Bailey. How she managed to get through that evening Polly never knew, yet it was not a long one, for at eight o'clock she slipped out of the room, meaning to run home alone and not compel anyone to serve as escort. But she did not succeed, for as she stood warming her rubbers at the dining room fire, wondering pensively as she did so if Maria Bailey had small feet, and if Tom ever put her rubbers on for her, the little overshoes were taken out of her hands, and Tom's voice said reproachfully, Did you really mean to run away and not let me go home with you? I'm not afraid, I didn't want to take you away, began Polly, secretly hoping she didn't look too pleased. But I like to be taken away. Why, it's a whole year since I went home with you, do you remember that? said Tom, slapping the rubbers about without any signs of haste. Does it seem long? Everlasting. Polly meant to say that quite easily and smiling credulously at his answer, but in spite of the coquettish little rose-colored hood she wore, in which she knew was very becoming, she did not look or speak gaily, and Tom saw something in the altered face that made him say hastily, I'm afraid you've been doing too much this winter, you look tired out, Polly. Oh no, it suits me to be very busy, and she began to drag on her gloves as if to prove it. But it doesn't suit me to have you get thin and pale, you know. Polly looked up to thank him, but never did, for there was something deeper than gratitude in the honest blue eyes that could not hide the truth entirely. Tom sighed, flushed all over his brown face, and dropping the rubbers with a crash took her hands, saying in his old impetuous way, Polly, I want to tell you something. Yes, I know, we've been expecting it. I hope you'll be very happy, Tom.  What! cried Tom, looking as if he thought she had lost her mind. Ned told us all about her. He thought it would be so, and when you spoke of another engagement, we knew you meant your own. But I didn't. Ned's the man. He told me to tell you. It's just settled. Is it Maria? cried Polly, holding onto a chair as if to be prepared for anything. Of course, who else should it be? He didn't say you talked about her most, and so we thought, stammered Polly falling into a sudden flutter. That I was in love? Well, I am, but not with her. Oh! and Polly caught her breath as if a dash of cold water had fallen on her, for the more in earnest Tom grew the blunter he became. Do you want to know the name of the girl I've loved for more than a year? Well, it's Polly. As he spoke, Tom stretched out his arms to her with a sort of mute eloquence that cannot be resisted, and Polly went straight into them without a word. Never mind what happened for a little bit. Love scenes, if genuine, are indescribable, for to those who have enacted them the most elaborate description seems tame, and to those who have not the simplest picture seems overdone. So romances had better let imagination paint for them that which is above all art, and leave their lovers to themselves during the happiest minutes of their lives. Before long Tom and Polly were sitting side by side, enjoying the blissful state of mind which usually follows the first step out of our work-a-day world into the glorified region wherein lovers rapturously exist for a month or two. Tom just sat and looked at Polly as if he found it difficult to believe that the winter of his discontent had ended in this glorious spring. But Polly, being a true woman, asked questions, even while she laughed and cried for joy. Now, Tom, how could I know you loved me when you went away and never said a word, she began, in a tenderly reproachful tone, thinking of the hard year she had spent? And how could I have the courage to say a word when I had nothing on the face of the earth to offer you but my worthless self? answered Tom warmly. That was all I wanted, whispered Polly, in a tone which caused him to feel that the race of angels was not entirely extinct. I've always been fond of you, my Polly, but I never realized how fond till just before I went away. I wasn't free, you know, and besides I had a strong impression that you liked Sidney in spite of the damper which Fan hinted you gave him last winter. He's such a capital fellow, I really don't see how you could help it.  You are so good to me those last days that I came very near speaking out but couldn't bear to seem to be offering you a poor, disgraced sort of fellow whom tricks wouldn't have and no one seemed to think worth much. No, I said to myself, Polly ought to have the best. If Sid can get her, let him, and I won't say a word. I'll try to be better worthy her friendship anyway, and perhaps when I've proved that I can do something and I'm not ashamed to work, then, if Polly is free, I shan't be afraid to try my chance. So I held my tongue, worked like a horse, satisfied myself and others that I could get my living honestly, and then came home to see if there was any hope for me. And I was waiting for you all the time, said a soft voice close to his shoulder, for Polly was much touched by Tom's manly efforts to deserve her. I didn't mean to do it the first minute, but look about me a little and be sure Sid was all right.  I couldn't wait another minute, or let you either, and I couldn't help stretching all my arms to my little wife, God bless her, though I know I don't deserve her. Tom's voice got lower and lower as he spoke, and his face was full of an emotion of which he need not be ashamed, for a very sincere love ennobled him, making him humble, where a shallower affection would have been proud of its success. Polly understood this and found the honest, hearty speech of her lover more eloquent than poetry itself. Her hand stole up to his cheek, and she leaned her own confidingly against the rough coat, as she said in her frank simple way. Tom, dear, don't say that, as if I was the best girl in the world. I've got ever so many faults, and I want you to know them all and help me cure them as you have your own. Waiting has not done us any harm, and I love you all the better for your trial. But I'm afraid your year has been harder than mine. You look so much older and graver than when you went away. You never would complain, but I've had a feeling that you were going through a good deal more than any of us guessed. Pretty tough work at first I own. It was also new and strange. I'm afraid I shouldn't have stood it if it had not been for Ned. He'd laugh and say poo if he heard me say it, but it's true, nevertheless, that he's a grandfellow and helped me through the first six months like a, well, a brother as he is. There was no reason why he should go out of his way to back up a shiftless party like me, yet he did, and made many things easy and safe that would have been confoundingly hard and dangerous if I'd been left to myself. The only way I can explain it is that it's a family trait, and as natural to the brother as it is to the sister. It's a shot trait to do the same, but tell me about Maria. Is Ned really engaged to her? Very much so.  He hadn't time to send by me. I came off in such a hurry. Maria is a sensible pretty girl, and Ned will be a happy old fellow. Why did you let us think it was you? I only teased fan a little. I did like Maria, for she reminded me of you sometimes, and was such a kind, cozy little woman I couldn't help enjoying her society after a hard day's work. But Ned got jealous, and then I knew he wasn't earnest, so I left him a clear field and promised not to breathe the word to anyone till he had got a yes or no from his Maria. I wish I'd known it, sighed Polly.  So they do, for neither you nor fan gave us poor fellows the least hint about Sid, and there I've been having all sorts of scares about you. Serves us right. Brothers and sisters shouldn't have secrets from each other. We never will again. Did you miss me very much? Yes, Tom, very, very much. My patient little Polly.  See if I didn't. And with great pride Tom produced a portly pocketbook stuffed with business-like documents of a most imposing appearance, opened a private compartment, and took out a worn-looking paper, unfolded it carefully, and displayed a small brown object which gave out a faint fragrance. That's the rose you put in the birthday cake, and next week we'll have a fresh one in another jolly little cake which you'll make me. You left it on the floor of my den the night we talked there, and I've kept it ever since. There's love and romance for you. Polly touched the little relic, treasured for a year, and smiled to read the words My Polly's Rose, scribbled under the crumbling leaves. I didn't know you could be so sentimental, she said, looking so pleased that he did not regret confessing his folly. I never was till I loved you, my dear, and I'm not very bad yet, for I don't wear my posy necks in my heart, but where I can see it every day, and so never forget for whom I am working. I shouldn't wonder if that bit of nonsense had kept me economical, honest, and hard at it, for I never opened my pocket-book that I didn't think of you. That's lovely, Tom, and Polly found it so touching that she felt for her handkerchief, but Tom took it away and made her laugh instead of cry by saying in a weedle-some tone, I don't believe you did as much for all your romance, did you now? If you won't laugh, I'll show you my treasures. I began first and I've worn them longest. As she spoke, Polly drew out the old blocket, opened it, and showed the picture Tom gave her in the bag of peanuts, cut small and fitted in on one side. On the other was a curl of reddish hair and a black button. How Tom laughed when he saw them. You don't mean you've kept that frightful guy of a boy all this time? Polly, Polly, you are the most faithful loveress, as Maude says, that was ever known. Don't flatter yourself that I've worn it all these years, sir. I only put it in last spring because I didn't dare to ask for one of the new ones. The button came off the old coat you insisted on wearing after the failure, as if it was your duty to look as shabby as possible in the curl I stole from Maude. Aren't we silly? He didn't seem to think so, and after a short pause for refreshments Polly turned serious and said anxiously, when must you go back to your hard work? In a week or two, but it won't seem drudgery now, for you'll write every day and I shall feel that I'm working to get a home for you. That will give me a forty-man power, and I'll pay up my debts and get a good start, and then Ned and I will be married and go into partnership, and we'll all be the happiest, busiest people in the West. It sounds delightful, but won't it take a long time, Tom? Only a few years, and we needn't wait a minute after SID is paid, if you don't mind beginning rather low down, Polly. I'd rather work up with you than SID idle while you toil away all alone. That's the way father and mother did, and I think they were very happy in spite of the poverty and hard work. Then we'll do it by another year, for I must get more salary before I take you away from a good home here. I wish, oh, Polly, how I wish I had half of the money I've wasted to make you comfortable now. Never mind, I don't want it.    She was as sure of that as if an oracle had foretold it, and was not deceived. For the loving heart that had always seen, believed, and tried to strengthen all good impulses in Tom was well repaid for its instinctive trust by the happiness of the years to come. Yes, she said hopefully, I know you will succeed, for the best thing a man can have is work with a purpose in it and the will to do it heartily. There is one other thing Polly, answered Tom, turning her face up a little that he might see his inspiration shining in her eyes. What is it, dear? A good woman to love and help him all his life, as you will me, please God. Even though she is old-fashioned, whispered Polly, with happy eyes, the brighter for their tears, as she looked up at the young man who, through her, had caught a glimpse of the truest success, and was not ashamed to owe it to love and labor, two beautiful old fashions that began long ago with the first pair in Eden. Lest any of my young readers who have honored Maude with their interest should suffer the pangs of unsatisfied curiosity as to her future, I will add for the benefit that she did not marry Will, but remained a busy, lively spinster all her days and kept house for her father in the most delightful manner. Will's ministerial dream came to pass in the course of time, however, and a gentle bright-eyed lady ruled over the parsonage whom the reverend William called his little Jane. Farther into futurity even this brash pen dares not proceed, but pauses here concluding in the words of the dear old fairy tales, and so they were married and all lived happily till they died.


What's up everyone it's your girl Kenzie here from Layton Sports Cards and I'm ripping a couple personals here for Colin G. We've got a 2022 Panini Prism FIFA World Cup Soccer first off the line hobby box as well as three 2122 Panini Prism basketball choice boxes. All right Colin good luck thanks for hopping in. It's gonna be a fun rip here. Let's start with the choice boxes. We've got Silver Rookie of Aaron Wiggins Thunder, Silver Lonnie Walker the fourth, our exclusive cards. Yeah we're walking them back here. We've got a Greenscope Patrick Beverly T. Wolves that is eight out of eights on the Greenscope and Red Scope Reggie Lewis for the Celtics 23 of 88s. Reggie Lewis. Sounds good Donald have a good day seeing a bit we had Joe Harris for the Nets, Danilo Galanari Hawks and we have a Bill Russell Celtics legend and your auto is going to be a Trey Murphy the third on the Red Scope rookie. New Orleans not numbered on that one. Right box two. Thank you. Thank you Donald. All right our Silvers Clay Thompson and Kyle Kuzma. Back here we have Red Clint Capello for the Hawks. That one is 24 out of 88. Love this blue color match. Blue Scope Terrence Ross for the Magic that is 30 out of 39. It's got a pretty big print line right here on the bottom across the Orlando Magic. Send a box here with you in case you want to take that up. Our choice exclusives. We've got a Rajan Rondo, Raja Rondo, Wakers, Rookie of Simonovic Bulls and a Scotty Barnes. There you go Colin. Nice Scotty Barnes on the choice exclusive. Blue, yellow, green. Nice looking card there. And our auto rockets throwback of Kenny Smith. Kenny Smith. No but he's just taken some time to himself Crumb.  All right box three. There we go. Last box before we get into your photo. Here Colin, we have Zach Levine Silver and Josh Richardson. Red Scope Mike Conley for the Jazz. And Sixers Andre Drummond. So here are your scopes. The Conley is 41 out of 88. And the blue scope is 26 out of 39. Lonnie Walker the fourth. Choice exclusive here. Robert Williams the third for the Celtics. And Rookie of Greg Brown the third for the Blazers. Playing video games all day. Probably. And we've got Clippers Red Scope Rookie of Keon Johnson. Clips. All right that'll do it for the prison boxes there. Now time for some photo Bifa. All right. Let's see a National Landmarks Manga or Color Wheel along with some as I call them shimmy shimmies. See some nice shimmies here. Some shimmy shimmies. Let's see it. Good luck Colin. Silver. We're Naldobies. We've got Pink. Christian Erickson. Nice one there for Denmark. Japan. People make big money streaming video games. They can make a lot of money doing that. There's different tournaments you can get into. And to make money we've got Alacer Johnson Silver for Canada. And so we've got Hyper of Sardar Azmoud for Iran. Oh Joe Allen. Phil Foden. Doku. We've got Red Mojo of Philip Kostich for Serbia. That one is 36 of 99. Animal Crossing Carl. Dude Animal Crossing is one of my favorite games of my life. Really? I wish I would have had somebody to play it with. Never played it. It is so fun. Never played it. Neymar. Puska. Joe Felix. Silver of Savic for Serbia. Messy Global Reach. Bolo. Nintendo sponsored the whole thing. Wow. I believe we've got our auto. A. Ferran Torres on the International Ink for Spain on the blue shimmer auto. It's nice looking autograph too. Nice one there. Not numbered on that. Got blue Luca Toria for Uruguay. 293 of 299. Silver Poland. Danny Omo. We have Cracked Ice Rookie of Koigo for Hasi for Japan. Not numbered on that. Cracked Ice. Roger E. Global Reach. Bellingham Base. Cam Silver Rookie of Al Hados for Qatar. Garcia. Oh geez. That's crazy girl. Junior. There's like seasons. Hyper of Empadu for Wales. It's like day and night and stuff. Collect all these different things. You would love it. It's all about collecting. They have this museum and you have to fill the museum up with all the things you've collected. We've got blue shimmer of Nicholas Gonzalez for Argentina. That is four out of five. There you go. World Cup. Argentina. World Cup. One of those nice ones there. Red and Matthew Ryan. That is 346 of 399. Connection Silver. That one's called US. Peppy and Buccio. Summer. Thank you Justin. A couple packs left here for you Colin. Sorry. Greenish. Silver of Marcello Florence for Mexico. Global Reach Wang. All right. Last pack. Got orange. Orange of we've got Sergei Nabri for Germany. 10 of 25. That is very nice. There is a bit of a dent in that right here. It's a little hard to see. Very nice hit though. Germany. We're going to mag this one for you again. A little bit of a dent there. So we're going to send the box here with you Colin. Yeah. A little bit of a dent on the top right there. Still going to mag it though and send the box with you. If you'd like to take that up. Penny. Scores Club. Zeus. And that will do it for you. All right. That will do it for you Colin. Thanks again for hopping in. Congrats on those hits so we'll get them right out to you.


What's up everybody, Jason here for Jasper's Case Breaks.com. 2022 Benini Obsidian Football just sold out. This is the second half of the case, six box break, number four. So here we go guys. So looking for some big hits in the second half guys, just like the first two breaks, one and two, the second half was really loaded. Didn't see any big case hits, no big RPAs, and we did get a couple golds and two colored glass in number two, so kind of expecting the same thing here. So hopefully we can get some nice hits for you guys. So thank you guys for the bottom to the R&B and of course everybody else at bottom straight up as well, without you guys, we wouldn't be able to do this break. So here we go. Yeah man, I think we do a great job as well man. I mean, obviously not trying to sound like cocking or anything like that, but I feel like the value out there for a lot of repack products, you know, we do pretty much give out a good value for the most parts though. I definitely agree. All right, here we go. Second half here guys. Hamilton for the Ravens, going to Victor straight up and out of 50, Trevor Lawrence Vitrius and that is Jacksonville Jaguars, part of the 10 team R&B, nine, which is Andrew. We've got colorblast, we do. Colorblast behind John and Taylor. Cooper Cup Colorblast for the Los Angeles Rams. Rams is Patrick Davis. There we go. Basically a cell sheet here right there. All right, then of course Colt's Rubber Runkel. Jonathan Taylor to 75. Another one there for the Colt's Rub. Yeah, obsidian's always been really nice, man. I love this in soccer. I thought I don't love basketball, but I love that they have colorblast and obsidian for those two products. All right, we have 43 out of 199, Trevor Penning. Saints, going to Joshua Santos. Aiden Hutchinson to 199, 74 out of 199 autograph. That is Lyons Robb, and then out of 175, Tryon Davis Price. That's Aliphanse. All right, next one. Yeah, baseball and this would have been awesome, but I just don't think they get enough traction and they probably figured they can't, but ooh, we got a white contra. So I don't know if we'll get a second colorblast guys, but we have a white contra coming up, which is considered another case hit as well. So behind Tom Brady, we have a white contra. It's Sam Howe, wow. That actually would be a pretty big time right there. Joshua Santos with that one. That's a super short print right there as well. And these white contra's are really rare as well. These are obviously probably more rare than colorblasts because there's only about, let's say 20 colorblasts. There's one of every player of this. So these are definitely, definitely rare. Very nice hit right there. And of course, Tom Brady, 98 out of 100. Nice one there for the box game. All right, Aiden Hutchinson. So there's a chance we don't get a second colorblast guys, but there's still gonna be some big hits though. 44 to 75, Lydani and Tomlinson for the chargers. Dan V put that one. We have a Kyren Williams for the Rams. Nice one there for the Rams, Patrick Davis. And we want Kenny Pickett, but I think Mike Vellucci, big Steelers fan, will take a big Ben Rafflesburg. Magnetic, future Hall of Famer. Very nice hit. And then Jalen Tolbert to 175 for the Cowboys. Going to add. Here we go, next one, this one's a thick pack. Deandre Swift and Lamar Jackson Purple. 72 out of 75, Ravens, going to victor to 100. All right, I feel like we haven't got any hits for the R&B yet, just see it right. Strange. Well actually, sorry, Jagsby did. That was Jeremy, just one. And here's another one. That's a non-number to one though, but everybody gets a shot at it. Collette Edwards, E-Layer, nice two-giller patch. Marcus Allen in the set, nice. I'd like to hit that for you. 45 out of 50, Kent City Chiefs, going to David. Tyreek Hill to 49. And Brody, I love this next card, man, but I know obviously it didn't really work out, although I had a feeling it wasn't really going to work out so much, more of a bridge. But Dolphins is part of the 10 team R&B, 25. Spot five, that's Ryan. Ryan Bellender. But still a nice hit though for your Falcons. 16 out of 49, looks like the little acetate top of it is coming off at the bottom, but Marcus Marriota, now on my Eagles, which my dream has come true. Eight years later, Marriota is finally an Eagle. And I think this is probably the best team he's going to be playing on by far out of his eight years in his career. So if he has a chance to play it because of injury, I hope he kills it. 75 out of 175, little sauce gardener patch autograph. And again, that's going to the Jets, Michael. Nice, no, yeah, I didn't check the checklist, but if he's in there, oh yeah. I've got Derwin James and a Supernova AJ Brown, one out of 25. Supernova AJ Brown in 99, that's part of the RMB. And that is 61, Peter with that one. Yeah, I think he fits perfectly with the offense, man, honestly. You know, I hope he doesn't play, because if he plays that most likely means either A, the Eagles are blowing him out of the team, so he is playing, but if he is playing, the other outcome is that hurts his injured or something, you know? So don't want that. But I think he's going to have a pretty bounce back season though, if he gets a chance to play due to injuries on the starter. You know, obviously with Atlanta, they didn't really have a good team. He showed flashes, but obviously, like I said, this is probably the best team he's been on his whole career. And I think he'll fit in perfectly there. All right, try on Davis Price. Yeah, I love them on the Raiders too. I obviously thought he had a chance to take over, but again, when he played for you guys, he looked great, broke off big runs, but then got injured, right? So it didn't really work out so much, but when he did play, it wasn't bad at all. Two out of 25, Russell Wilson, Broncos Scott. I remember that one game during COVID actually, when he played against Justin Herbert, when the car got injured, that game was sweet. Actually almost won that game too. Three out of 75, Fasan Haskins, Tennessee Titans, 10 team R&B, three. Three is going to Jeremy Muller. And there we go. How about a Kenny Pickett 18 out of 50?  There you go, man. All right, you guys, two more boxes. Oh, yeah, Wobby Big Oregon fans, so that's why I love them a lot, but yeah, he was pretty good, man. All right, we have a Tyler Smith to 50, and how about a little Herbie? Orbital. Nice one, 82 out of 99. Pierre Strong Jr. for the Patriots, that's part of the 10 team R&B. That's another non-numbered cargo randomized. Travis Etn to 125 for the Jaguars. That's a 10 team R&B, 91. One is Peter. Matt Corral to 75. That's another one there for the R&B. 14, Spuff Forest Trevor. Oh, a little Reggie Wayne. Reggie Wayne, Aurora. Aurora, sorry, 31. That's Robert Rungo with the Colts, though. And then out of 175 with the Detroit Lions, Jameson Williams and Robert G with that one. All right, guys, here we go, lost box. Rob Chuck, how you doing, buddy? All right, Bailey Zappi, nice one there to 25. And an A.J. Brown tunnel vision. Eagles part of the 10 team R&B, 68, 68. Yeah, I think Detroit's heading in the right direction, man, honestly. 10 team R&B, love for Zappi, 14, Spuff Forest Trevor. We got Devontae Smith, another one there, not numbered. So those three not numbered cargoes, we're gonna randomize them in one lot. We have a Chris Godwin, number to 125. Nice one for the Bugscape. George Pickens to 50. Two color patch, another one for the Steelers by Greg Gallucci. Jonathan Taylor, lightning strike for the Colts. Nice one there for Rob. And last hit of the break, Matt Corralto, 175. Got a patch auto. That's for the Carolina Panthers, part of the 10 team R&B. And it's 125, Spuff 5, Ryan B. And there you go, guys. So that was the break, very solid hits popping out of here. We did get a color blast with just Cooper Cup.  And they're getting Aiden Hutchinson-Auto, Ben Rothesberger, Saus Gardner, Kenny Pickett, months other big names. That was pretty nice. So there we go, guys. So we did have three not numbered cards, guys. That'll be going to the R&B, everything else was numbered. But we'll randomize those in one lot. Name it number one, get to all those cards in one lot. So I appreciate you guys, the switch scenes. All right, sorry about that, Ryan. All right, roll it. One out of four or five times, good luck. One, two, three, four, five.  So there you go, Trevor. Appreciate it, guys. Coming up next is a little filler, and then we'll do some F1, and then some honors down the night. Appreciate it, guys.


So, welcome to chapter 2 of the course. I have named it as creating new spaces. We shall discuss some standard procedures of constructing new topological spaces out of the given ones as well as constructing totally new ones also. Of course, we shall not just construct them, we shall also put them in proper prospective, study them later more. That is all this chapter is about. We begin with two fundamental concepts. These concepts apart from helping the topological study elsewhere, give immediate methods of constructing all topological spaces in a general fashion. Some simple examples that we have considered in the previous chapter in an ad hoc fashion will now become motivating examples for some systematic studies. I will point out to them whenever we come across those things. So, welcome to module 22, Basis and Subbasis. You may as well take one of them today. We will see, but I am already laying foundation for the second one also. So, today maybe we will just discuss basis because lack of time. So, the first observation is that start with any set and a family of topologies on X. Take the intersection of all these families. Which means what? All members of Px, may be subsets of X, which are in every member of this family, tau j's. So, there is an intersection. That family will become a topology on X. So, this is what I meant by saying that creating new topologies out of old topologies for example, this comes into that category. Tau j's are a families of topologies. Now, I am taking the intersection of them. Perhaps this one is a new one. It is not always. If you take say tau 1 contains a tau 2, then this intersection will be just tau 1. So, it is not always the new one. That is why it is a general method. So, this size of topology after all is very important concept for us. The proof is very easy. Let me just see what we have to do. Empty set and the whole set is inside psi. Why? Because it is there inside every topology.  So, their intersection will be in each tau j. So, intersection is here. Similarly, if you have a family of open sets as u alpha, members of psi, that means what? Each tau j contains all of u alpha. Therefore, the union of u alpha is inside tau j for every j. So, that means that union of y is a member of psi. So, that completes the proof of this topology. This is just like in a vector space. If you have a family of vector subspaces, then the intersection is a vector space. Similarly, if you have a family of subgroups of a group, then the intersection is a subgroup. However, in either of the cases, the union utterly fails to be a subgroup here and some space there, vector some space there. The same thing is happening here. If you take two topologies, the union may not be a topology. Of course, if one is contained in the other, union is the other one. So, that is a very special case. In general, union may not be a topology. But we do not want to give up just like that. So, we will work harder to get topologies out of these things also. So, what we will do? So, here is another general method, which is actually an application of our previous theorem here. So, this theorem is a foundational. It is like it keeps helping us again and again. So, start with any family of subsets. Instead of taking topologies and so on, just any family of subsets of x. In the collection of all topologies on x, which contain this s, there is a unique topology tau s, which is the smallest. First of all, the discrete topology, the entire px contains this one. So, this family of all topologies is non-empty. Logically, it is not necessary, but we can fix it up that it is non-empty. Now, I am taking intersection of all these topologies, which contain s. This theorem says that such intersection is a topology. And since all of them contain s, this will contain s also. So, one part is fine. Namely, we have got a topology which contains s now. But this is the smallest one. Why? Because it is contained in the each of them because this being the intersection of all of them. So, that is all. So, one step application of this one gives you this result. Namely, given any set, there is a smallest topology. That is a unique smallest topology. It is the intersection of all topologies containing that, unique one. Why unique one? Because there are two of them, the intersection, intersection must be equal to both of them. So, we can make a definition. What is that? The topology tau s, namely the smallest topology containing s, is called the topology generated by s. If s itself is a topology, what happens? Tau s will be equal to s because s is already a topology. So, tau s and everything else contains it. So, intersection will be s. So, if s is already a topology, no problem. But if s is not a topology, then this is. So, this is again similar to what we do in linear algebra. You have a vector space. You have a set of vectors there. How do you create a vector subspace which will contain all these points, all these vectors? So, that is the vector space generated by this. So, same word we have taken here. The same terminology. So, this one step consequence of our first theorem is going to be extremely useful method of creating new topologies because I may start with any subset of the power, any subset of x, any collection from the power set that will give you a unique topology, the smallest one that contains that. Of course, what may happen is two different sets, two different subsets of x may give you the subsets of p x I am taking, collection of subsets of x may give you same topology that is possible. Taoist may be Taoist brand. So, here is a simple exercise. Take x equal to z or any countable infinite set and take s equal to x minus one single point, n belong to z. One point is missing from each subset here. Look at that is the collection. What is the Tao s corresponding to this s? What is the topology generated by this family on the set of integers? It turns out to be something familiar to you. Can you see what is happening? Okay, I do not mind. This is precisely my point, namely we need to work harder to identify the topology generated by s s s. Like in the case of vector spaces, we give a collection of sets, collection of elements. What are all the elements in the vector space generated by that? They will be finite linear combinations of elements from this set. So, that is a description. So, we would like to have similar description here. So, let us try that. So, in order to be able to identify Taoist, identify members of Taoist, it is necessary to establish or invent a method to describe the elements of the topology generated by s in terms of the sets inside s. Okay, so here again, you see the motivation comes from matrix spaces. So, that is why certain things there we have done just to add up fashion perhaps, whatever motivated them there, but now they will motivate to enter new constructions here. Recall that a metric space x d, okay, open walls where basic objects in define the topology for Tao. It is Tao d, right? Open walls, right? That is a good example that will guide you, okay. So, how was the Tao d defined then arbitrary union of open walls, all right? So, based on that, we will do something now, namely we will define another term here, base for a topology. So, that was the title of the today's topic after all. A subset of px that means the collection of subsets of x is called a base for a topology on x if the following conditions are satisfied. First condition is that if you take union of members of b, that should cover the whole of x. Second condition is that given two members of b and a point inside the intersection, you must have a third member in b, okay, such that x is inside b3 contained in b1 intersection b2. When I say third member, it is I am not saying that they are all distinct, no. But the point is neither b1 intersection b2 itself may not be there, that is the point, okay. If b1 is contained in b2, intersection will be just b2, it is just b1 and so on. So, those are easy cases. So, you may have two different subsets, they may not intersect. So, then also I do not have any moderation, but if they intersect for every point inside b intersection b2, I must have a member of b such that that member is contained in the intersection and contains the point, okay. So, this is pay attention to this one, this may take little more time. But look at the set of open balls in a matrix place. This is precisely what we had proved there. If you take intersection of two open balls, it need not be an open ball. But every point inside it contains an open ball, every point is around that point that is a open ball contained in the intersection, okay. So, that is why we have proved this condition. So, that is motivation is from open balls inside a matrix place. Once b1 and b2 are satisfied, look at the topology generated by b now, tau b, that is what the least topology which contains b. Then we say that this b is a base for tau, okay. Now, here is a theorem. First of all, to characterize what kind of collections will be a base for some topology, okay. So, start with a collection b and let tau be any topology. Will b be a base for tau? This is the question we want to address now. So, this theorem says that the following three conditions are equivalent. The first condition is b is a base for tau, that is our motivation. The second condition is that first of all b is contained inside tau which is very obvious if b is going to be a base for tau and every member of tau can be expressed as a union of members of b. So, this is precisely the result that we proved for open balls. In fact, sorry, the definition that we had for topology from the open balls, okay. For tau b was precisely defined this way, right. So, that condition comes here. Third condition is of course, b contained inside tau is common given any x inside u where u is inside tau. tau is what? tau is the given topology. So, u is an open subset in this topology. I must find a b inside b such that x belongs to b contained inside u. These three conditions are equivalent in the statement of this theorem. Let us go through this proof. One implies two, okay. If b is a topology, b is a base for tau, okay, by the definition tau has to contain inside, contain b. So, that is for the second part is, second part is what? That every member of tau can be expressed as union of members of b. Okay. So, take tau prime with the family of all subsets of x which can be expressed as a union of members of b, okay. Let this be the family. Then I want to show that this tau prime itself is a topology. The proof is exactly similar to what we did for the open balls. There we had to work harder there. Here we have made whatever that work into a definition here, namely the condition, okay. So, let us see tau prime be the family of all subsets of x which can be expressed as union of members of b, okay. So, why tau prime is a topology? Again, the axiom Au, Au is what arbitrary union is easy because union of union, union over the families of union is again another union. To see the finite intersection, suppose you verify it for two of them, then again the intersection of unions will be unions of intersections. Therefore, it is enough to verify it for only two of them anyway. But I have written complete proof here, U equal to union of Vi, V equal to union of VJs, okay. Let them be members of tau prime, that is by definition. What are UIVJs? They are inside b. Now, we look at intersection, it is Uij, okay. Intersection taken over all i and j, Ui intersection Vj. Now, suppose x is inside the intersection, U intersection V left hand side, then it must be inside one of these, okay. This means Ui intersection Uj contains x for some ij because the union is a union. So, there exists a b belong to b, this is the part of the definition for b is a base. So, that x belongs to b contained inside U intersection V, Uj, Ui intersection Vj. It is a part of the definition for b is a base. So, that is what I am using here.  Therefore, U intersection V is a union of members of b, every point is inside b. So, it will be union. So, therefore, it is an element of tau prime. Once this is a family, this family is a topology, the s tau b, which is the smallest one will be contained inside tau prime. So, that is every member of tau can be expressed as union of members of b. Once it is contained inside tau prime, what is tau prime? It is expressed, every member is expressed as union of members of b, okay. So, this proves one implies two. Now, two implies three. That is very straightforward because one, every member expressed as union of members of b, even any x belong to U, given b is contained inside tau is already there.  So, this part is very easy. Once again, three implies one will be the hardest at all, if I tau, even that is not difficult. So, let us go through that. In addition to proving conditions b1 and b2, we must also prove that tau is the smallest topology containing b. So, this is the meaning of b is a base for tau. First of all, I should condition b1 and b2 should be this way. Then the smallest topology containing b must be equal to tau. So, these two things I have to show, actually three things I have to show, okay. Since every member of x belongs to some member of b, okay, b1 holds. So, this part was starting with what we had, three implies, three implies one is what I have to show, right. Given any x belong to U inside tau, there exists a b belong to b, tau, members of tau cover all of x, right. So, x belong to U, then there is a b belong to U which contains that one. So, union of all members of b must be cover the whole of x. That part comes from here, okay. So, that is first part here, okay. That also I have to observe. So, b1 holds. Secondly, b is contained inside tau, that is given, right. In three, that is given, okay. b is contained inside tau. Given b1 and b2 belong to b, intersection will be member of tau because each of them is a member of tau. You take U equal to b1 intersection b2 and apply this condition 3 for U, b1 intersection b2. You get a b, such that x belong to be contained in the b1 intersection, b2 now in the intersection, okay. So, b2 also holds. Finally, I have to show that tau b is equal to the given tau here, okay. b is contained inside tau. Therefore, tau b is contained inside tau because tau b is the smallest topology. I have to show that tau is contained inside tau b, okay. Only that one remains now here. Okay. So, we want to show that tau prime, any topology on x such that b is contained inside tau prime, okay. Then we want to show that this tau is contained inside tau prime. That will show that tau prime, tau is the smallest topology containing b, that is tau b, tau, right. So, given any element in tau, for each x inside U, we have some bx belonging to b, such that bx is contained inside U. This is given to us. Therefore, U is the union of bx, x belong to U, right. Note that each bx belongs to b but they are all contained, b is contained inside tau prime. Therefore, and tau prime is a topology. Therefore, the union of all bx must be in tau prime. So, starting with U which in element of tau, I have shown that it is in tau prime. That is all, okay. So, tau is the smallest topology containing b, that means tau b is equal to tau. So, equivalence of 1, 2, 3 is established by that. So, that is what I am telling here which I have told already. If x d is a metric space and b is the collection of all open balls in it, clearly this b satisfies b1 and we have proved that it satisfies b2 also in that theorem, chapter 1. Therefore, it is a base for a topology and that topology is nothing but tau d as much we have seen anyway. In particular, the collection of all open intervals in R is a base for the usual topology. I just want to recall all these things. They were the basic things, they were the motivating examples for us, okay. For the future, okay, the purpose is future. Right now, we do not need it because there is this b1 occurring. So, we can make that as a separate definition, namely any collection U of subsets of x satisfying the condition b1 is called a cover for x, okay. Union of members of U, if it is equal to whole of x, then we say U is a cover for x. This we will use later on, okay. So, what we have seen above, okay, we cannot expect any arbitrary cover of b to generate a topology. We need condition b2 also, right. This b2 may not be satisfied by the given b. However, there is one type of families which satisfy this property, okay. It is stronger than b2. That is why I want to pay, I want to call your attention to this one, all right. We will take up that one next time and that will lead to the concept of sub-basis. So, thank you. So, we will meet next time.