Welcome to module 17 of point set topology course. This time we shall consider a few more interesting examples of topological spaces. The most interesting one that we are going to do today will be a subspace of r itself. Before that, let me consider something which is out of r or not, not the usual topology. Start with any infinite set, then I am denoting this topology COF, CO-finite, so you can read it as co-finite. So what is this? It is collection of all subsets A of x such that the complement of A is finite. Of course, I will have to allow A to be empty also because complement of empty set is the whole space, I do not want that, in this definition I do not have that one, x minus A is finite. If x is, if A is x, of course, it is allowed, A is empty is not allowed in this rule, so I have to allow it specifically, so either A is empty or x minus A is finite, then I put it inside this collection, co-finite topology. So I want to say that with this it becomes a topology, ok. So this is called the co-finite topology on the given set x. Once you consider such a topology, it is no point in considering x to be a finite set. Then what happens? This becomes just a discrete because all subsets will be there because their complements are also finite. So it is interesting only when you take x to be an infinite set, ok. How to verify that this is a topology that is not very difficult because what you have to do? Once you take two of them x minus A1 and x minus A2, right, their intersection, what is the complement of intersection? It is the union of the complements, right. So each of them is finite, so union is also finite. So intersection of two open sets is open or intersection of finitely many open sets is open. The union of arbitrary families is open because as soon as one of them complement is finite, if you enlarge it larger and larger, say it will be automatically finite. So that part is easier here. So that will verify that this co-finite becomes a, you know, a family of co-finites subsets, it becomes a topology. We will have many, many instances of, you know, usefulness of this one throughout the course, ok. So this is going to be kind of model for us. One of the interesting property, you can directly verify it right now, namely take any point in x, look at all the neighbourhoods of this point. Intersection is precisely the single written x, that is the only point x is common to all the neighbourhoods, ok. So verify the statement. Just to begin with, getting familiarity with this co-finite topology, we will have many instances of studying this one. So this is just an introduction right now. Similar to this one, but not so important is the following, ok. Take x to be an uncountable set, let COC is in a co-countable, denote the set of all subsets such that either it is empty, that empty set is allowed or the complement is countable. Now countable includes finiteness as well as infinite countable also, ok. So countable means that, alright. So once again verification that this topology is identical to the previous one, there is no problem, ok. Because countable, finite union of countable sets is countable, that is all I am using here, ok. So this is called the co-countable topologies, just similar to co-finite topology, ok. So it has similar properties. Once again, there is no point in taking this topology when x is a countable set. Then again it will be a discrete topology, therefore I take x to be starting with an uncountable set, ok. Let us go to the third example here, actually there are two of them. This time I come to the underlying set is R itself but topology is different. So I better write it in its different symbols. So here I have used LR respectively RR to denote collection of all intervals of the form minus infinity to A to denote that these are left-handed rays, LR, R for ray instead of arbitrary these are the intervals but to indicate that they are unbounded, unbounded intervals, right. Unbounded below, no. So LR, the other one is RR of the form A to infinity. A varies to minus infinity to A, this is LR, A to infinity there RR, right rays, ok. Where this A ray, A can be taken as all the way from minus infinity included here. So this one, what does it mean? Empty set. When it means minus infinity here, this will be empty set. When it is press infinity, open interval I am taking remember that. So that will be just the whole of R, ok. So this forms a topology because if you take intersection of minus infinity A with minus infinity B, look at whether A is B bigger than B or B is bigger than A, the smaller one intersection will be just smaller one. If you take some A alphas here and take the union, then the union will be what? Take the supreme of all these A alphas and take that open interval for minus infinity at one. This exactly similar thing will hold for this also. Here you may have to take infimum to show that arbitrary union of open rays is again an open ray. So this is even simpler than the topology in which all open intervals are allowed, right? The real line. So obviously in both of them there are fewer open sets than in the usual topology on R. So they are different topologies in any case. For example, the open ray here to the right will not be an open subset in the LR, see, and vice versa, all right? So these things are again important in analysis. So when an opportunity arises, I will again refer to this one. Now I come to what I told as the most important example. This time I have a subset of R itself with the usual topology, okay? So this is called Cantor set, which is a landmark result in topology, okay? A landmark result by Cantor. First we shall define an operator. You can read it as Cantor. I don't know how to read this symbol. Here I read it as Cantor. On the class of all closed intervals A, B, A less than B, okay? I am not interested in singleton Bs. So A, B, A less than B, the closed intervals like that. Into the class of all closed and bounded subsets of R, okay? So what is this C? C is going to be an operator. It takes a closed interval and then produces a closed and bounded subset of the same interval A, B, okay? So prior to that we define another simpler operator free on the class of finite union of disjoint closed intervals, which you may call middle one-third radiators, okay? So this funny name, we will explain it right now. Start with any closed interval. It may denote it as J, okay? So A, B. Let us define phi of J to be the set obtained by deleting the open part of the one-third of the middle one-third, okay? Open part. So in notations what is it?  So delete that from J. So what you get is disjoint union of two closed intervals. So that is the definition of phi J. Take any closed interval, 0, 1, A to B, 5 to 15, whatever. So it will produce two disjoint closed interval, each of length one-third of the original one, right? From the starting point to that one-third distance and then two-third distance to the end, okay? Very simple operator. The middle one-third is deleted. That is why it is called middle one-third deleted, okay? Now you extend phi to the union of finitely many closed intervals, disjoint closed intervals, okay? So A is union of disjoint closed intervals. Define phi A to be union of phi of A, I, B, I. So here I have defined phi of A, B. So that is why you have defined phi of A, I, B, I. Take the union of this one. They will be again disjoint because each A, I, B, I is contained inside A. Each phi of A, I, B, I is contained inside A, I, B, I. And they are disjoint here, okay? So this is just extending from one interval to finitely many intervals. So phi A, definition of phi A is over. Now what we do? Now fix one interval A, B, okay? Let us call it I, naught. Inductively define, see I, one will be what? Phi of I, naught. That means the first I delete, the first one middle one-third of A, B. Now I have got two intervals there. Operate phi on that one, okay? So that is what I have to do. Okay, phi of I, one will have two intervals, right? Take that as phi I, two and so on. So I, n will be phi of I, n minus 1. So when you come to nth level, there will be how many intervals? Two power n intervals will be there, they are disjoint. All of them will be contained in the previous sets here. So this is a strictly monotonically decreasing sequence of closed and bounded sets, right? Strictly monotonically increasing sequence. What I am going to do, finally the canter of A, B is your operator. It is just take the intersection of all this. It is like taking the limit, limit of sets, but limits of sets is defined very easily. It just take intersection of, okay? This is definite. One thing is very clear that because all these are closed subsets, this will close. So this is a closed and bounded subset. Starting with an interval, I have produced a closed and bounded subset of A, B itself. I notice A, B and all these are subsets of that, okay? So here is a picture what I have done. Start with one interval, delete the middle one-third. So you have two intervals here. Now look at this one, delete the middle one-third. So this will give you two here, two here. So at the second level all day you have four of them. Now delete one-third here, delete one-third here and here also. So the third level you will have eight of them and so on. So keep going on so on. So you will get a lot of just, things looking just like dot, dot, dot, dots, okay? So that the limiting thing is the Cantor set, okay? The function C is called the Cantor's construction, okay? The set, now I am taking specialization. Instead of A, B put zero one, standard one. Take the closed interval zero one, apply the Cantor operation. Let us denote that by C, okay? Usually one can, one usually verifies Cantor set means that one, but there are more wider applicable meaning for this one, okay? Right now what is the meaning of Cantor set? When take A, B equal to zero one and apply C, okay? The sets C of A, B, so whenever there is arbitrary A, B I will write like this. When there is just zero one, I will just write C, that is all, okay? Have some wonderful properties, okay? So let me list a few of them. Not all these properties may be easy for you. Certain things may need some concepts which you do not know, but this is actually a topic analysis. So I will just list them right now. I may not explain all of them to you, okay? Quite a few of them I am doing it. They are easier, so or I will explain them right now, okay? The first thing is C of A, B is non-empty. I already told you it is closed and bounded. That is easy. Why it is non-empty? So this is Cantor's intersection theorem, okay? So the first thing is very important here. It is a non-trivial thing, but this is non-empty. If J is one of the intervals contained in I n. So what I mean by say, look at this picture. This is I naught, I 1, I 2, I 3. When one of I n, look at one of these intervals, the small intervals, okay? The intervals I am talking about, not arbitrary intervals. It is not subset of these intervals, but one of these intervals. You just take that. So if J is one of the intervals contained in I n for some n, then if you look at C of J, that would be also subset of C of A, B. You can just apply C of that because C is an operator operating on any intervals, closed intervals. So C of J is C of A, B because J is contained in A. Given any, the endpoints are always inside C of A, B. See, endpoints are never deleted. These two endpoints is endpoints point, okay? So beauty is to begin with I can say these are the endpoints. They are not deleted. But now this point and this point will never get deleted because from here I could have started and arrived at one. So this point, the endpoints are never deleted. The same thing is here. Endpoints, these endpoints are never deleted. So in the end, these endpoints are all there. They are never deleted. If something is somewhere in between, you never know whether it will get deleted. So that is the beauty of this. So there are all endpoints of successively cutting intervals. They are all there. That is the whole idea. Though I say it only for A, B, because it is true for all subintervals also, okay? Now take the map fx equal to A plus B minus A times x. It is a linear map, okay? What does it do? Put x equal to 0, you will get A. x equal to 1, you will get B. So that is a linear map, isomorphism, homeomorphism from 0, 1 to AB, right? It just expands. This one-third becomes B minus A by 3, right? So what happens is the cantor set C of AB and cantor set C of 01. This is C. They will be isomorphic under this continuous bijection. So this map preserves the cantor construction. It is, first of all, map defined from 0, 1, 3. Whatever middle one-third deleted, the middle one-third correspondingly will be deleted. This is not an isometry because 01 is expanded to AB, right? But it is a similarity. For now onwards, I will just concentrate on the cantor set C of 01. Each of the properties of C which we list below is carried over to an identical or similar property of C of AB, right? Because of similarity map. So that is why I do not have to say it for all of them. Once I say it for C, it will prove for C of AB for all of them. So here are some more properties of C. These are special in the sense that they are all points between 01. That is the speciality. Topologically, when you have, when you have similarity, they may not be inside AB, 01, but maybe it will be inside AB. That is the difference. The end points of every component of IN, n greater than or 0, they are inside this wall. We have, we have observed already. The set of all rationales of the form 1 to n, okay, the finite sum, ak divided by 3 power k, okay? What are ak? Ak is either 0 or 2. That is contained inside C. So why this happens? Look at the first time, the very first interval here, okay? Everything from here to here is one-third. From here to here, one-third plus something, okay? So that point from one-third plus something, that will never be there. From here to here, it is one-third to third plus something, that will be always there. So 2 will be there. From here to here, it is just within, it is in one-third. So the first will be 0. 0 plus something by 1 by 9, it may be. And plus something by something, I mean something by 1 by 27. So into some 27 and so on. So that is what this means. The ternary expansion is what is being done. I am writing every element as 1 by 3 or 0 by 3 plus something a by 9 plus something b by 27 and so on, right? So there is a sequence ak. These ak's will never be equal to 1, okay? They are either 0 or 2, okay? So this is an easy consequence of the construction here. The middle one-third is deleted, okay? All the rationales of this form, okay? They will be there, alright? Now C contains no open intervals. There is a first observation, okay, that I am making which is now serious. Take any open interval, okay? So first of all, it should be inside AB, inside 01, alright? So it is somewhere here. What happens? Go on making one-third, 1 by 9 and so on. Finally, some small portion will lie between the same interval because interval has positive length. And that one-third portion will get deleted, okay? So that is easy to see. So no open interval will be contained inside canter set in the final stage, okay? In between there, there are there. These are intervals. But at the end, there are no open intervals, okay? What is the meaning of that? Interior is empty. It is a closed set already. Therefore, it is no air dense, you see? So suddenly we have constructed no air dense set here, okay? So where are we? Here, okay? So C contains no open intervals. In particular, being a closed set, it is no air dense. Otherwise I have to take a closure and then look at the interior. So closure of C is C itself. Therefore, interior of that is empty. So this means it is no air dense. Every point of C, the limit point of C. Limit point means cluster point. Is that very difficult to see? Take a point of C, it is a limit point of C. Take an interval, okay? There will be some other point other than that point. That is all I have to see, right? So no point will be isolated, okay? So for this kind of thing, you can just keep glaring at this construction one by one. You can produce that. Or you can use your arithmetic and compute things and so on. Just look at every element of this form, okay? So near that, you can take any epsilon. You can produce one more element other than this one, okay? So that kind of arithmetic will produce proofs also. So it is not all this. So what is it? Every point is a cluster point. Every point is a limit point, okay? Such sets are called perfect sets. So L of C is equal to C. The fifth one here, the fifth one is C is uncountable. So this follows by looking at all these distinct, you know, suppose you have a sequence here, okay? And take infinite sequence also. See, this I have not said here. Look at all infinite sequences. The entries are either 0 or 2. Take a sequence and then conform this sum. This will be a convergent sequence, okay? Right? If you truncate it as finite level, all these points are there. The limit is also there because it is a closed set, right? So infinite sequences of this form 0, 2, 0, 2, whatever give you a number here in the cantile set. But these sequences are themselves uncountable, okay? Now all that you have to see is that if you take two different sequences, they always give you different numbers, okay? This is similar to what you do with the static expansions. So here we are doing ternary expansion. You could have done it with any number here, okay? With the decimal expansion also you could have done, okay? If it is one-third, there is a charm here which is not there with some other number. So we will see what come to that one later. Okay? So where were we? Uncountable. C is of length 0. Now you may not know what is the meaning of length of this arbitrary subsets here, right? If you do not know that, don't worry about that. I will tell you what it is whenever it is needed. The third, the last point here I am going to make is given any two points inside C, okay? So assume X is less than Y. There exists disjoint clause of states A, B of C such that C is A union B, A and B are disjoint clause. X is inside A and Y is inside B. So such a thing is called a separation. Whenever any two points can be separated like this, okay? Such a set is called, such a topological space is called totally disconnected, okay? So don't worry about this one. This is a very profound topological property which we will discuss only in part two. But this property I have very nicely defined and this can be proved right directly. There is no problem for the canter set, okay? For the canter set, we will prove this one, all right? So that is not very difficult. So in any case, I have put all these proofs here also. Let me do the last one first and then come back to all of them. So look at this one. Take any interval. This interval is not contained inside C, which we have observed earlier. So take a point x, say z between x and y, strictly between x and y, which is not in C. If it is not in C, what happens? Take the closed interval 0 z, intersect with C. See, this is a closed subset of R. This is a closed subset of R. Intersection is a closed subset. Similarly, B equal to z to 1 and intersect with C, okay? z to 1 contains y, right? So B will be inside y, sorry, y is inside B. Similarly, x will be inside A. And by the very definition, the intersection of this will be just z, but z is not there. So intersection is empty, okay? So both A, B are closed in C, x is inside A, y inside B. So this is a proof for last k, okay? Now this is what I am telling you. What is the meaning of length is 0, okay? What I am subtracting? For the first time, for the first time, I am doing 1 third, right? Then I am again subtracting 1 by 9, right? So this j, I am trying to approve this j. C is of length 0, okay? You sum up all the lengths, okay? That summation is equal to 1.   So if some number is less than epsilon for every epsilon, that must be 0. So that is the meaning of this. This totality of total length of all the intervals that I am going to subtract. First, this is 1 third. Then from whatever remaining, I am taking 1 third here and 1 third here. 1 third here, 1 third here. 1 third, 1 third, 1 third, 1 third. 1 third of the original thing, right? So I keep taking. So this is 1 third. This will be 1 by 9, 1 by 9, 2 by 9. This will be 1 by 27, 1 by 27, like 4 by 27. So you see nothing but 2 power n divided by 3 power n. Take the summation. That is what it is here. 2 power n minus 1 divided by 3 power n. Some total is 1. So whatever remaining is 1 minus 1. So length will be 0. So I think I will turn down all these proofs here. More carefully, you can go through that. You will get the nodes in a case, okay? So y interior is empty. I have explained it completely here. So this canter set is going to play a lot of interesting role in analysis and also in topology, okay? So once again, let me give you some exercises here. All right? Start with a equal to 1 by n. n belong to n. Include 0. So compute all these things under the... This is when old XI remain under topology R, under topology is different topology. Co-finite topology, co-countable topology and LR topology. The original topology you have done it was... Now you do it for these three topologies. Once you do it for LR, it is similar to RR. So it won't be much difficult for you. That is similar. Okay? So do that exercise. Here there is another exercise. Convergent sequences and dense subsets. You have described in co-finite topology and co-countable topology. Okay? All this doesn't need many more education. You have been given the definition. Definition of what is a convergent sequence. Definition of what is the meaning of dense subsets and so on. You have to work out for them for here. Okay? A sequence converges to a point if given any neighborhood, there must be some n such that k bigger than n means Xk is inside that neighborhood. So that is the kind of definition. But now you apply to cleverly chosen neighborhoods of points inside co-finite topology and see what happens. Similarly for this one. See what happens. Okay? So that will be the material for today. Let us stop here. Thank you.


The Equitable Society presents this is your FBI. This is your FBI, an official broadcast from the files of the Federal Bureau of Investigation presented as a public service by the Equitable Life Assurance Society of the United States. To your FBI, you look for national security and to the Equitable Society for Financial Security. These two great institutions are dedicated to the protection of you, your home, and your country. Tonight's file, The Desert Dictator. There is an ancient proverb which says, The sword of the wicked shall break against the wisdom of the righteous. History bears witness to the inexorable truth of that proverb. And likewise does tonight's case from the files of your FBI, in which a ruthless criminal, a killer, is defeated by it. On a desert in Arizona, two elderly men and a girl are seated before a campfire watching the red sun toys for a moment over the peak of a distant mountain. Gradually, the red turns to pink, then to purple, then indigo. Now, it is night. Doc? Sorry, freshen up that fire. You could see a little better. See better? Yeah. He didn't hear a word you said, Mesa. No. No, Larry. Uncle Harold, you know that when you're making notes, you get sowing growth. You could do it in the dark. You exaggerate by talent. You've got company. Wolf. Yeah. Well, he won't bother us. I imagine wolves are particularly interested in archaeology. Doc? Yeah? I'd like to ask you something. Go ahead. I like you, and I like your niece here. You're paying me good money to guide you around the desert. Well, just what satisfaction do you get out of digging stuff out the ground that's been there for hundreds, thousands of years? Well, that's a fair question. Let me answer it, Uncle Harold. Sure. You see, Mesa, the past can be a guide to the present. It's important to know what made one period of civilization die in order that this one shall survive. In other words, Mesa, what Larry means is if Hitler had used archaeology instead of mythology, he would have known better. Well, that makes sense, all right. Here, Mesa, the compiled data on the three ancient Indian cultures that followed each other in this country. The first period was called basket maker number one. See, at that point, the Indians were nomads, hunters. Then basket maker number two. What's wrong? Just the horses, Larry. That wolf may be flirting around with them. I'm going to take a look. Well, what's the matter with you critters? What you got your noses in the wind about? Settle down, no. Settle down. They are now, Miss Larry. I don't know, but something has. What do you mean? I got a feeling that we ain't by ourselves out here. What? What's that?  I said drop it. Now look, I'm in the dark and you're in a light that makes you a real easy target. I ain't dropping no gun. I'll leave it lay there. The next one slams into you. Are you all right, Miss? Yes, ma'am. The Wild West boys ain't the only ones that can shoot, you know. Who are you? This company. Well, you're not exactly welcome. That ain't what I'm after. His arm is bleeding. He's been bleeding all day. Does it worry you? No. Thanks. What's the setup here? Well, speak up when you talk. I'm an archeologist. Do you know what that means? I know what it means. Mummies. Well, that would be the common Egyptologist. Never mind, lad. Where do you fit in the picture, sweetheart? Me? Yeah. This old ghee, your husband? He is my uncle. Bet you couldn't help. Who's Buffalo Bill here? Our guide. I thought maybe it doesn't matter the ground. Now just a minute.  Any bandage, sweetheart? Bandage. For my arm. Go get it. When I'm fixed up, I'll tell all of you what you've got to do to be able to see the next sunrise. Okay? A few minutes before closing time that afternoon, a lone bandit had held up the Canyon National Bank and Flagstaff, Arizona, and escaped with $20,000 after a running gun battle. The FBI office in Phoenix was notified. And shortly after sundown, special agent Blake drove into Flagstaff where he conferred at once with Sheriff Hickman. Two of my boys saw the bandit when he dashed out of the bank and jumped into his car. And they opened fire on him? Well, they yelled at him to stop first, but he had his car started by then when turned off. The boys fired a couple of shots, come in there to car and took after him. How did he make his getaway, Sheriff? Well, he managed to shoot out a tire. Was he hit at all, you know? The boys weren't sure about that. But once after they fired, they saw his car sware a little. Which way was he heading? He was aced on the road leading through the painted desert. Sheriff, did you get a description of him at all? Yes, yes. I got it written down here for you. I've already phoned all the Sheriff's between here and New Mexico. Oh, good. He was driving a black Buick sedan. Here's the license number. California. No, probably stolen. Well, we'll put out a three-state alarm right away. And after we check with California, we'll start fanning out men all over the state. All right. But if he gets off in that desert country, it'll be hard to find. Well, he can't stay in there forever. Sheriff, hand me the phone, will you? We'll get that alarm out. There you are. Thanks, sweetheart. Yeah, it's a real good vantage. You put it on with love and care. I put it on because I had to.  He's evidently evolved a theory, Larry, that you're really fascinated with. Never mind a book talk, Doug. Let me give you the real feeling on me. Please do. I suppose you've all guessed by now that I didn't get shot cleaning a pistol. I borrowed the 20,000 bucks that's in this bag from a bank today.  Our car? That's right. I didn't waltz two miles across this desert to your campfire just to get a little first aid. My car burned out of bearing. I traded in for yours, except that I pushed it over a cliff. What's so funny? You're in a bad way, unless you can use a horse. What do you mean? We haven't got a car. You're lying. Your car on this camp somewhere. You're at liberty to look for yourself. Of course, if you care to wait until tomorrow. Larry, what did you start to say about tomorrow? I was going to say the sheriff, or whoever's after you would probably be here by then. Be very glad to give you a lift. You were talking about something else where he wouldn't have stopped you. What was it? I just told you. What was it, sweetheart? Come on, tell me. Wait, miss. I'll tell him. Okay. There's somebody else in your party, right? Everybody else in our party. Okay. But somebody's going to be here with a car tomorrow. Yes. We're expecting supplies. I'm sorry, Uncle Harry. Don't be sorry, sweetheart. That helped. I was beginning to figure I was in a pretty bad spot. Well, since you think you see your way out of this now, perhaps you won't mind if we all retire. Sure, go ahead. But you'll have to hit the ground right here by the fire. Oh, yes, of course. In dark. Yes. In case you think you got an edge because you went to college, I just got out of San Quentin University last week. Apparently, I didn't teach you much. I taught myself, Doc. I taught myself how to stay up three days and nights without sleep. Oh. Yeah. So go ahead with the sleeper night. But don't count on me passing out. I'll be sitting right here waiting to say good morning to all of you. And a very special good morning to whoever's coming with that car. Morning, Sheriff. Good morning, Mr. Blake. Sheriff, this is Special Agent Tanner. Hello, Sheriff. How you doing, Tanner? I just talked on the phone today. Los Angeles Police. That black-viewing sedan the bandit Jews were stolen there three days ago. We know who the bandit is now, Sheriff. Oh, yes. Tanner here checked your description of him with our Washington office. His name is Matt Ricker. He just got out of San Quentin last week. Well, gone back to work again already, huh? Here's his full description. 5 feet 10, 165 pounds, black hair, dark skin, has a knife scar on his right cheek. Want me to put that out? We've already sent it out as a follow-up on your first alarm. Sheriff, have you had any reports from any of your men? No, sir. But by now, they're pretty well spread out over this part of the state. Oh, good. Tanner and I are just getting ready to take a run across the painted desert road to Gallup. If we get anything, communicate with the sheriff there. We'll be in touch with him. And we'll double back here to Flagstaff. All right. Good. Come on, Tanner. Let's get started. Well, how did everybody sleep? How do you think? You should have knocked it off real good knowing somebody was standing by watching over you. That was very comforting. Thanks. Hey, Pop, pour me another cup of that... What are you doing there? Just fixing the fire. Why? You're lying. You're doing tricks with the smoke trying to make a signal. All right. Get away from there before I slam that in you. You pour me a cup of that Java. I ain't waiting on you. I said pour me a cup of that Java. I'll get it for you. Pop's going to get it for me. I ain't getting you nothing. Okay, Pop. You asked for it. No, don't. Wait a minute. Listen. Oh, this is what I've been waiting for. All right. Just stay put.  There'll be some blood on his sand. Morning, Marshall. How's everything going? Just fine, Mr. Marshall. I got a good mess of supplies for you. Is that so? Yeah. Wow. Looks like you took on a new member since I was out here last, Doctor. That's right. Who is he? He just joined us last night. That's funny. What's funny? You look quite a bit like a man I heard about yesterday. A man they're looking for back in Flagstaff. And if I'm not mistaken, his name is Matt Rodgers. Oh, that's it. It didn't help you, Ricker. What do you mean? Look at the car. Your bullet pierced the gas tank. Instead of shooting your way to freedom, you've destroyed your only avenue of escape. We momentarily close the Equitable Society's presentation of the Federal Bureau of Investigation File on the Desert Dictator. We'll return to this case in just a moment. Tonight, I'd like to tell you a tale of two cities and 12 letters. One city is New York, where the Home Office of the Equitable Life Assurance Society of the United States is located. The other city we'll call Your Town, a prosperous manufacturing community somewhere in the United States. Now, on a certain day, 12 letters were exchanged between Your Town and the Equitable Society. Nine came from equitable members living in Your Town, enclosing payments on their Equitable Life Insurance policies. On the same day, three letters from the Equitable Society, also enclosing checks, arrived in Your Town. One check was for a very large sum of money, an investment of Equitable Society funds in Your Town's most important industry. A second check went to a respected doctor who was retired under the Equitable Old Age Security Plan. The third check went to a young man in Your State University, a payment on the educational insurance his father provided for him years ago. Well, that's one way that life insurance works. Money comes from Your Town to the Equitable Society. Money goes back to Your Town from the Equitable Society. Now, in place of Your Town, substitute any town, city, or village in the United States in which some of the Equitable Society's three-and-a-quarter million members live, and then multiply those 12 letters into several million a year. And it becomes very clear why we say that by serving its members, the Equitable Society serves America. Now, back to the file on the Desert Dictator. The cynical irony of quicksand is that the more desperately one struggles to escape from it, the deeper one sinks into it. Matt Ricker feels the grim touch of this same irony as he watches the gasoline flow from a bullet hole in the tank of the automobile in which he had hoped to escape from the desert and the vicinity of his crime of bank robbery. All right, so I blew that chance for getting out of here. Don't waste any breath laughing at me because I'm not worried a bit. You'd hardly laugh at the body of an innocent man lying there. You've got a lot more to answer for now, Mr. You don't think I'm going to sit here and wait for the Lord to come and get me, do you? Ricker, I wasn't very interested in you before, but now I am. What do you mean? But to this point, you've just been a petty annoyance that we've been helpless to remove. From where I stand, you're still helpless, Dave. That remains to be seen. What do you think you're going to do? That depends entirely on what you do. No matter what I do, the yards are all in my favor as far as you're concerned. They're all in favor of the rabbit, too. But he didn't win the race. Look, Doc, I got two more days and nights before I even have to think about sleeping. Now, let's get moving. In a particular direction? I'm going to put that up to Buffalo Bill here. But what up to me?  What if I do? You're going to take us out of this desert to a good hideout over in the mountains somewhere. I'll figure out the rest when we get there. I ain't taking you nowhere. This time is running out. I'd like for us to be there when it does. Whatever you say, Doctor. We'll pack up and let's get moving. Oh, maybe this will get a word out of you, sweetheart. Well... I'll have to ride double with somebody. Guess who I've picked. Thanks. That way I won't have to look at you. Stop playing hard to get. Okay, let's get out of here. Failing to pick up any trace of liquor on their run to Gallup across the desert, Special Agent Blake and Tanner of the FBI started back to Flagstaff. And late that afternoon as they drove along the desert road, Blake saw something to the left of the road where a ground broke sharply off into a shallow canyon. Uh-oh. What's that? What do you see, Bob? I didn't notice that when we passed you this morning. What? Yeah. Look back there where the ground drops are. Wow. Looks like car tracks. Yeah. Come on, let's take a look. The car turned off the road here all right. The tracks go right out to the edge there, too. Yeah. Hey, Bob, look down below. It's Ricker's car. Let's get down there. Oh, wait a minute. Ricker won't be in it. He's heavy footprints in the middle of the tire tracks. He must have pushed it over the side himself. Yeah. I guess it either broke down or he decided it was time to make other arrangements. Look, these footprints go across the road toward the desert. Yeah. Yeah, there they go. That's not mistaken. Ricker was wounded. Look. See those little dark spots here? There's some more here. Hey, I just happened to think. You remember that archaeologist we met in town a couple of weeks ago? Dr. Endicott? Yeah, that's him. Well, from what he told us, I believe this camp is right out in this direction. Hey, that's right. And if Ricker were here last night, he probably saw and it gets campfire and made for it. He could have seen the fire from miles. We better get out of that camp. Where's that trail start? A few miles up the road. Come on. Isn't that Tom Marshall's car there? Yes. Endicott seems to have broken camp. There's something wrong about this, Bob. What are you thinking? Hold it. It's Marshall. This must be Ricker's work. He's evidently taken Endicott's party with him. There are no supplies here. Well, then he's trying to get somewhere to hide for a while. Come on. It's getting dark fast. Look around for any sign of what direction they took. Here are some hoof prints. A lot of them. Leading north. Well, there's nothing but open country to the north. Well, maybe the trail turns later in... Hey, wait. Find something? Yeah, give me a flashlight. I hear you. Now, look at that. Looks like somebody's been drawing pictures in the sand. Well, those are Indian symbols. Well, that's your dish, Bob. What do you make of it? Well, I'd say Dr. Endicott left him. That first sign is a snake. Ricker. Yeah. The stair steps. The stair step pyramid represents a mountain. The sun going down behind it means west. Yeah, but what's that twisting line across the mountain? Well, that's the symbol for a trail or a road. Come on, let's get Marshall's body back to the trading post and call Flagstaff. I want to get a map of this region. You know, this is real nice. Smells good, doesn't it? With you around, Mr. Ricker, it's difficult to tell. I get it, Doc. Very funny. How about you, sweetheart? Don't that moon kind of hit you? I have to agree with my uncle. Thanks. Hey, wait a minute, Pop. I told you not to fool around with that fire. I'm just putting on some more wood. And none of that signal stuff. I said I'm just putting on wood. Okay. You know something? These mountains really get me. The guy I knew in San Quentin used to make big talk about a country like this. He'd talk about a row of trees like there were. Beautiful Farley's Dames or something. You know, I know what he means. How did he feel about killing people? What do you mean? How did he feel about shooting someone in cold blood as you did today? That's why he was in San Quentin. Let's turn in, Larry. Hold it. You guys can turn in.  What? I want someone to help me look at that moon. Come on, sweetheart. Let's take a little walk. With you? Yeah. You're gonna be sorry you've done that. All of you are gonna be sorry.  The car. The car? The car way up here. I forgot to tell you there's a road that runs 50 feet west of here. What about that fire? They're a little too late, Mr. Laker. Oh, no, I hate them. Give me that gun. Let go of me. All right, Laker. Drop that gun. Drop it. Okay. Thank you, Mr. Blake. I gather you found the symbols. Yes, Dr. Endicott. There's one you neglected to draw, sir. The electric chair. Matt Ricker was tried and convicted on the charge of first-degree murder. He was put to death in the state penitentiary. The career of crime is a losing game. And neither the criminal's wit nor his cunning or his guns can overcome it. For they are no match for the thing which inevitably defeats him.  He is but one. They are many. He is fallible. They are sure-footed. He is wrong. They are right. And as the ancient proverb says, the sword of the wicked shall break against the wisdom of the righteous. You'll hear about next week's case in just a minute. This week at the Equitable Society, Thomas I. Parkinson, president of the society, showed me a letter he'd received from an American mother who was mighty proud of her son. He's just been released from the Army Air Corps, holds two decorations, and is a veteran of 47 bombing missions. In her letter to President Parkinson, this mother said, I think the members of the Equitable Society can be proud of my son too, because the Equitable Society helped his father and me plan his education. The Equitable Society helped me keep my home after his father died and bring my son up as his father wanted me to. Well, you can imagine how happy that letter made us all. For no matter how often it happens, we're always glad to see what a warm, friendly feeling equitable members have for their society. Also, that letter reminded us that years ago, when that mother's son was still a baby, the Equitable Society's management was just as alert to the needs of the future as the Equitable Management is today. It made us happy to know that as we work hard today to keep the Equitable Society progressive and forward-looking, we're following a great tradition. Yes, this week and every week for over 86 years, the Equitable Society has been building security for you, your home, and your country. Next week, we will bring you another colorful story from the files of the Federal Bureau of Investigation, the bogus bankruptcy. The incidents used in tonight's Equitable Society's broadcast are taken from the files of the Federal Bureau of Investigation. However, all names used are fictitious, and any similarity thereof to the names of persons living or dead is accidental. Programs in this series of particular interest to servicemen and women are broadcast overseas through the worldwide facilities of the Armed Forces Radio Service. Tonight, the music was under the direction of Frederick Steiner, the author was Frank Ferries, and your narrator was Dean Carlton. This is your FBI is a jerry divine production. And now this is Carl Frank speaking for the Equitable Life Assurance Society of the United States and inviting you to tune in again next week at this same time for this is your FBI. This is the American Broadcasting Company.


two pieces of kind of cool news this morning. One, the shark's lights are over there. The motor was making a terrible noise since the last time I fixed it. It was just because I tightened something too tight so I put some shims in. Anyway, fixed it. Now it sounds super smooth. So much better. The other thing is I have a very good lead on being able to cover the sails for this thing with solar panels. Like the whole thing with high quality solar panels. It'll cost me a bit of money but not like something I can afford. Oh, I'll say more about that later. So these solar panels, I don't want to say too much because it's not a done deal. However, I think I just need to go answer the email I got this morning and say yes. And then it's a done deal I think. Alright, let's dry. Yellow. Of course, I can do some touch-ups with a paintbrush. First, I'm gonna wait to see if I like it. You know, when this has the cover on it, this is gonna look like a snail. I mean it already looks like a snail, a slug. But then it's gonna have this big round thing back here like a snail shell. Maybe I should paint it like a snail shell. That thing there. I wonder if I can put my covers on yet. I don't know. I think they might still stick. But I don't want this to get rained in more. It got a little bit of water in yesterday. Alright, that should keep the rain out of it. These ones fit loose so I just gently rested them on there. Now the paddle wheel cover, which goes right here, I want to paint maps on it. Like maps of the local area. Like right here, a nice map of between here and town and then maybe some dotted-out routes with all the little obstacles in the way, the shallow areas, the mangrove islands, everything I have to go through. Now I just need to give this a few days to harden. I guess in the meantime I can go work on my island. Oh, back to digging. Alright. Oh chair, you look so so cute. I found this floaty carpet out in the water like, I don't know, a year ago. And I think I can make a chair thing if I, yeah. I also made a new hoodie because I wanted something that could keep me out of the sun. I feel like I can pedal for hours in this thing now. Oh, that's gonna be good. I kind of like build some sunglasses into my hood here or something. This thing fit right between the two metal things here. No problem. Everything else went back together, no problem. And this may be just me making things up, but I'm pretty sure the boat's faster now because I, you know, the bottom of that pontoon I was working on is smoother now than I was before. So theoretically it should definitely be faster. And I feel like it's faster, but I always feel like it's faster when I just made something new, obviously. I put a stronger stick on the steering here, one of these things. So I don't have to worry about the little PVC pipe I had on there flexing when I'm trying to steer hard. Yeah, I'm really happy with this. I do still need to do some kind of hinge to hold these covers on. I want to, I want to look at some hinges I have, have this big strip of stainless steel hinge. I was gonna put that on, but I just don't think that's right. I'm gonna have to come up with something else. I think I need to make something fancy because it doesn't have a flat edge anywhere. So I'd only be able to put the hinge on a little spot. It wouldn't be that strong. I'll figure something out. I mean one thing I could do is just drill a hole through this right through the the lip that comes up in here, put a string through, and then it would just kind of be loose so you could take it off. And it wouldn't fall in the water or anything. Just put it on, but it would be loose. That's not a bad idea. I'm still thinking about it though. Alright, let's take it out for a spin. Man, that is one comfy seat. I think I'm going to make the exact same thing over here. This is just temporary one. I have a couple cushions in here for the temporary seat, but yeah, I need to make one really good like that. Inside the hatches are hopefully, yep, totally dry. Reigned a whole bunch this morning and yesterday. Good, I'm really happy with that. That's a cumulative knowledge of my pedal boat experience put into practice so far. Hopefully, I'll learn some cool stuff on this and be able to do even better next time. Man, this is pretty good. I'm really happy with the paddle wheel. You know, my old pedal boat, I didn't really optimize the paddle wheel. I was just like, I just need something that'll get me around, but this boat actually tried to make it pretty efficient and I think I got it pretty good. Stainless steel tools. I don't have a stainless axe in there yet.


Anger. Anger can be expressed constructively or destructively. Narcissistic rage, for example, is a destructive form of anger. But anger has very many important functions. It's actually a very positive thing. It fosters communication. It brings about compromise and behavioural change. It's a change agent. It affects our conduct with each other. Interactions, a social fabric. Today we are going to discuss the difference between anger and rage. How to express anger constructively and how the narcissist fails in doing so. My name is Sam Vaknin. I'm the author of Malignan Self-Love, Narcissism Revisited. And I'm a professor of psychology. There are three elements to expressing anger constructively. Follow this recipe and your anger may have favourable outcomes. Will not lead to hostility or conflict, but on the very contrary, will make your life a lot better. So three simple rules. Number one, communicate your anger honestly. Don't be coy. Don't be shy. Don't be underhanded. Don't be passive aggressive. Don't be embarrassed. Don't be socially inhibited or constricted about your anger. Express it honestly, firmly, but not aggressively.  Never internalize your anger. Never act in ways which express anger without verbalizing it. Always talk about your anger. Always speak about your anger. Describe it, dissect it, communicate it, pass it forward. Make sure other parties realize what is your state of mind. And this leads to the second element, the second piece of advice. Remember we are discussing constructive anger. The kind of anger that transforms your environment in ways which are self-efficacious, conducive to your aims, helpful. The first element, communicate your anger honestly. Never bottle it up. Never internalize it. Never suppress it. Never deny it. The second element is describe your state of mind. Don't blame. Don't guilt-trip. Don't shame. Don't attack. Don't aggress. Don't point fingers. Just describe your state of mind. Say, I'm feeling uncomfortable right now. I'm angry right now. And I would like to describe to you how it feels. It feels bad. I'm unhappy and sad. And there is something you can do about it. You can help me. I need help. Describe your state of mind. Ignore the others. Even the source of frustration. Even someone who had attacked you. Even an enemy, perceived enemy. Someone who is hostile to you, unfriendly. Let it go. Don't focus on other people. It's a hopeless task and it puts their defenses up and the situation becomes only worse. Focus on yourself. Communicate your inner state. Let other people know.  You can even describe your background, your personal history. Other events which tie up with this one are the provocations, problems you've had. Humanize yourself. Make yourself accessible. Share experiences so that you create commonality and evoke, hopefully, some empathy. Angry people are disempathic. When we are angry, our empathy level goes down. Try to restore it. Try to restore it by showing the other party that you are vulnerable and you are not ashamed to admit it by describing your vulnerability in great detail and by seeking help. That's the third element. Ask for change. So the first element, communicate your anger honestly. The second element, describe your state of mind. Don't focus on other people. And the third element, ask for change. Ask people around you to act differently. Act differently. Behave differently. Talk differently. New actions. Different behaviors. Novel speech acts. Ask people to change themselves. To transform themselves. So as to improve your state of mind and to reduce your anger. Sometimes people would say, I'm going to change if you're going to change. It's a trade-off. Offer change. Offer change in return if need be. Be assertive. Not submissive. Not aggressive. Assertive. In a minute I will talk about it. But anger to be constructive must induce change in the sources and agents of frustration. Anger must transform your environment. Otherwise it's futile. Otherwise it's violence. Otherwise it leads to more anger. More wounding. More injuries. Much worse situation. Communicate honestly. Describe your state of mind. Ask for change. Do it assertively. But what's the difference between assertiveness and aggressiveness? Reactance. Defiance. People confuse these things. They think that to be assertive is to be in your face uncompromising, rigid, principled. That's not true. Assertiveness and defiance should not be confused. Certiveness is the opposite of aggressiveness. Weak, insecure people are aggressive. People who are sure-footed. People who have high self-confidence and self-esteem. Whose sense of self-worth is regulated from the inside. People who have an internal office of control. They are assertive. They are not aggressive. Assertive people are self-aware. They know themselves well. Defiant people have little self-acquaintance. Assertive people have clear and firm boundaries. This is where I stop. This is where you start. Defiant people, aggressive people constantly test the boundaries of others, pushing as far as they can get without incurring a backlash. Assertive people discuss topics. Defiant people attack personalities. Assertive people are for anything and everything that's positive, constructive and leads the way forward. Defiant people are against everything. They are contrarian. They are negativistic. Assertive people collaborate. They are goal-oriented. They focus on accomplishing things. Defiant and aggressive people network aggressively and compete. They are focused on winning. They usually, at the expense of others, zero-sum game. Assertive people are self-constructive. They aspire to and they attain personal growth and development. Aggressive people are self-destructive and self-defeating. And that's precisely the difference between anger and rage, especially narcissistic rage. Narcissistic rage is a phenomenon that is unique to narcissists. It resembles very much a temper tantrum by a spoiled brat. And that's simply because narcissists are spoiled brats emotionally and mentally. They are fixated at age two. The lucky ones go up to age nine. But they are children.  They react to every provocation with rage. And narcissists, especially psychopathic narcissists, try to present themselves as imperturbable, resilient to stress, sang-fois. Narcissistic rage is not a reaction to stress. It is a reaction to a perceived slight, an insult, a criticism or disagreement. In other words, narcissistic rage is a reaction to narcissistic injury or narcissistic wound. Narcissistic rage is intense. It's disproportional to the so-called offense. The rage is verbally stalked by the narcissist as the episode progresses. So as opposed to anger, which usually reduces in intensity as time passes, rage increases in intensity. The rage is dissociative. Like the narcissist wakes up after the episode of narcissistic rage, often embarrassed and shame-faced and oblivious to many of his misbehaviors. That's especially true for narcissists who are also borderline. The rage never leads to any real-life consequences. It yields no decisions. It alters no long-term behavior patterns or commitments. And it affects no pre-rage emotions or cognitions. The narcissist remains curiously untouched by the tantrum, as though nothing had happened. One critical tactic, one common tactic of the narcissist is to project his pain. To project his pain and to give it to you, to hand it over to you. Similarly, he hands over his rage to you. Consider, for example, the shared fantasy. The shared fantasy is a concept invented by Sander in 1989. It is a process, not an event. It involves various stages, starting with love-bombing and grooming. When the shared fantasy is disrupted, it results in frustration and frustration leads to aggression. That's dolled 1939. A disrupted shared fantasy usually results in narcissistic injury or even worse, modification, and this leads to narcissistic rage. Now, shared fantasy can be disrupted in various ways and at every possible stage of the fantasy. So it can be disrupted, for example, via the infidelity of the partner, triangulation. It can be disrupted in the love-bombing or grooming phase. It can be disrupted by bargaining at any phase. Bargaining is when the partner insists on commitment and investment. Whenever the shared fantasy is disrupted, the narcissist experiences anticipatory abandonment anxiety, impotent rage, a sense of victimhood, profound sadness, and trepidation. And this is all very traumatic and modifying. But the narcissist cannot admit that he is experiencing pain and sadness and rage, emotions that are dysregulated out of his control. The narcissist is always in control, is godlike, so he cannot admit to experiencing these emotions. So what he does, he projects these emotions. He slices them off, he takes them as a package, and he hands them over to his intimate partner, to you. He defensively projects his ego-discrepant agony.  All these emotions threaten to dysregulate him, threaten to transform him into a borderline. He can't have that. He can't have that because it challenges the narcissist's grandiosity. The narcissist cannot admit that anything is more powerful than him, even his own emotions. This is ego-discrepancy. It sits ill, it sits badly. With his self-image or self-perception, his ego-ideal. So what the narcissist does, he hands this negative, this negative affectivity. He hands over, he transmits these negative emotions to you. And he tries to provoke in you past traumas, your own victimhood, helplessness and desolation. And what he says to himself is, what I'm experiencing right now is not my pain over me. It's not sadness because of me. I'm not angry because of me. It is my pain and sadness and anger over her, over my intimate partner.  I'm empathizing with her. I'm identifying with her. And I'm experiencing her pain because I love her and I care for her. This projection, this misattribution of emotions to another person, it's called attribution error. This helps the narcissist preserve his grandiose self-image and self-perception as impervious, invulnerable, invincible, unbreakable, victorious. He needs to feel in control at all times. So now he handed over his negative emotions to you. And he has to force you to affirm and confirm the counterfactual projection.  You are really experiencing these emotions. You have to lie to him in effect. You have to agree with his projection. You have to say, you're not said, I am said. You're angry and you're right to be angry because I was the one who started it. I was the one who was angry at first and so on. So you need to affirm this projection. The narcissist wants you to experience your pain, which is actually his pain that he had shared with you. To preserve his ego-syntony, to avoid cognitive dissonance, the narcissist lies to himself and says, I'm doing all this because I'm a good caring partner. I'm doing this for your own good. Tough love, you know. I'm just trying to heal you, just trying to improve you, just trying to help you. So this is one mechanism of expressing rage and experiencing anger vicariously through a third party, by proxy through you, which is a very poisonous, very toxic experience, because you keep absorbing all the narcissist's negativity. But this is in a shared fantasy. Usually outside the shared fantasy, the narcissist is much more overt, much more open. Raging narcissists usually perceive their reaction to have been triggered by an intentional provocation, this kind of paranoid ideation of a secretary delusion. They think people are hostile, malicious, malevolent, conspiring and colluding to hurt them. They think that there's a hostile response, a hostile purpose, I'm sorry, behind the provocation. Their targets, on the other hand, invariably regard-raging narcissists as incoherent and just an arbitrary. Narcissistic rage should not be confused with anger, though they have many things in common. It is not clear whether action diminishes anger or anger is used up in action, but anger in healthy people is diminished through action, through expression. That's why I advocate the honest communication of anger. When you communicate anger, it dissipates, it dissipates and disappears. Anger is an aversive, unpleasant emotion. It is intended to generate action in order to reduce frustration. Anger is coupled with physiological arousal and physiological arousal can go on forever. Another enigma is, do we become angry because we say that we are angry? Do we identify the anger and capture it by saying it aloud, speaking anger's name, does it evoke anger? Or do we say that we are angry because we are angry to begin with? Are we describing a phenomenon, an objective phenomenon, ontological, or are we mislabelling something? Anger is provoked by adverse treatment, deliberately or unintentionally inflicted. Such treatment, such mistreatment, must violate either prevailing conventions regarding social interactions, or some otherwise deeply ingrained sense of what is fair and what is just. Anger is intimately connected with a sense of justice. The judgment of fairness or justice is a cognitive function. Now the narcissist doesn't have this. He doesn't have this because he doesn't believe in justice or fairness. He believes in the law of the jungle and he believes that he is far superior to other people and therefore entitled to special treatment and dispensations and a special set of rules. Anger is induced by numerous factors. It is almost a universal reaction. Any threat to one's welfare, physical welfare, emotional, social, financial, mental, any threat to one's well-being is met with anger. It's a self-defense measure. Threats to one's affiliates, nearest, dearest nation, favorite football club, pet, pets, you name it. Any threat provokes anger. The territory of anger includes not only the angry person himself, but also his real and perceived environment and his social milieu. Threats are not the only situations to incite anger though. Anger is also the reaction to injustice, perceived injustice, real injustice. It's a reaction to disagreements. It's a reaction to inconvenience, discomfort caused by dysfunction. But still, all manner of angry people, narcissists or not narcissists, all angry people suffer from a cognitive deficit. They are worried. They are anxious. They are unable to conceptualize, to design effective strategies and to execute these strategies. Angry people dedicate all their attention to the here and now. They ignore the future consequences of their actions. In a way, they become a bit psychopathic. Their empathy is reduced. Recent events are judged to be more relevant and they are weighted more heavily than earlier events. Anger impairs cognition and reality testing, including the proper perception of space and time. In all people, narcissists and normal people, anger is associated with a suspension of empathy. Irritated people cannot empathize. Actually, counter empathy develops in a state of aggravated anger. The faculties of judgment, risk evaluation, they're altered by anger. Later provocative acts are judged to be more serious than earlier acts, just by virtue of their chronological position. Normal anger results in taking some action regarding the source of frustration, or at least the planning or contemplation of such an action. In contrast, pathological narcissistic rage is mostly directed at oneself or is displaced or even lacks a target altogether. We call it diffuse rage. Narcissists often vent their anger at insignificant people. They yell at a waitress. They berate an Uber driver. They publicly chide an underling. These are the wrong targets. The rage is displaced. It's a kind of passive-aggressive reaction, dishonest communication. Alternatively, narcissists sulk. They feel unhedonic or pathologically bored. They drink. They do drugs. They abuse substances. These are all forms of self-directed aggression. From time to time, they are no longer able to pretend or to suppress their rage. Even covert narcissists have it out with the real source of their anger. Then the narcissist, overt or covert, loses all vestiges of self-control, raised like a lunatic. The narcissist shouts incoherently, makes absurd accusations, distorts the facts, and airs long-suppressed grievances, allegations, and suspicions. These episodes are followed by periods of saccharine sentimentality and excessive flattering and submissiveness towards the victim of the latest rage attack. Driven by the mortal fear of being abandoned or ignored, the narcissist repulsively debases and demeans himself. That's another difference between healthy anger and narcissistic rage. Healthy anger does not impact the sense of self-worth, and it is never followed by any acts of submission and phoning or flattery. Healthy anger is stable. It diminishes and disappears once the environment had changed and actions had been taken. Narcissistic rage is abrupt. It's abrupt, and then it gives place to ego-destiny. It gives place to negative affectivities such as fear, anxiety, and in rare cases, shame and guilt. Most narcissists are prone to be angry. Their anger is always sudden, raging, frightening, and without an apparent provocation by an outside agent. It would seem that narcissists are in a constant state of rage, which is effectively controlled most of the time, but not always. The rage manifests itself only when the narcissist's defenses are down, when he decompensates, when he is incapacitated, when he is modified, when he is adversely affected by circumstances, internal or external. Pathological anger is neither coherent nor externally induced. It emanates from the inside. It is diffuse. It is directed at the world and at injustice in general. The narcissist is capable of identifying the immediate cause of his fury. And still, upon closer scrutiny, the cause that the narcissist has identified is likely to be found lacking. The anger is always excessive, disproportionate and incoherent. The narrative is disrupted and disruptive. It might be more accurate to say that the narcissist is expressing and experiencing two layers of anger simultaneously and always. The first layer of superficial ire is indeed directed at an identified target, the alleged cause of the eruption. But there's a second layer. The second layer incorporates the narcissist's self-aimed, self-directed wrath, self-loathing, self-hatred, self-punitive, sadistic aggression, inner-critical superego. Two layers. The narcissist's rage also has two forms. One is explosive. The narcissist flares up, attacks everyone in his immediate vicinity, causes damage to objects and people, and is verbally and psychologically abusing. The second type of narcissistic rage is pernicious or passive-aggressive. The narcissist sulks. He gives the silent treatment and is plotting how to punish the transgressor and put her in her proper place. These narcissists, mainly covert, are vindictive and often become stalkers. They harass and haunt the objects of their frustration. They sabotage and damage the work and possessions of people who they regard to be the sources of their mounting rage. As I mentioned in 1939, the American psychologist John Dollard and four of his colleagues put forth their famous frustration-aggression hypothesis. With minor modifications, it fits well the phenomenon of narcissistic rage. Stage one, the narcissist is frustrated in his pursuit of narcissistic supply. He is ignored, ridiculed, doubted, criticized. Phase two, frustration causes narcissistic injury or even modification. Phase three, the narcissist projects the bed-object onto the source of his frustration. He devalues her, attributes to her malice and other negative traits and behaviors. And finally, this causes the narcissist to rage against the perceived evil entity, the secretary object, that had so injured and frustrated him. As you can see, narcissistic rage has extremely little to do with anger. It's an internal process, externalized improperly and dysfunctionally.