 Thanks, Zach and Sarah, for giving me the chance to be here. Hopefully everybody can see my screen and hear me here. Okay, good. Thumbs up means good things. And thanks all of you for coming to the talk. I really appreciate that, especially we're after a break here. And so I used to have classes where there was a break sometimes and about half the class would leave after the break was over. So it's nice that everybody came back. So thank you. I'm going to talk about characterizations of two homeomorphic spaces, which is part of my thesis research. So first of all, let me get here. Wait, going too far. So let me give you a couple of blanket assumptions. I'm going to talk about taking off spaces in here, taking off is completely regular and house dwarf. Just a reminder, completely regular just means that the space has got enough continuous functions to separate points from. Close sets. So if you have a close at a, and you got a point that's not in a, there's some continuous function that's one at the point and then zero on the close set. It turns out you could talk about more general kinds of spaces here, but there's a reason why I focus on the ticket on spaces. I'll mention that a few slides from now. A few other little notational things. We use ordinals once in a while. So all ordinals have the order topology. So we can think of mega is the smallest infinite ordinal. So that's just the natural numbers and the order of topology on that turns out to be just the usual discreet topology. So we can think of mega as the naturals or just as some sequence without any limit point attached to it. And if we got two spaces that are homeomorphic, we'll just use the approximately equals to side indicate that. Okay. All right, so let me tell you what a, what a two homeomorphism is. I think the definition is pretty recent. This is a 2018 paper by Arkin. Jelsky and maxi, you know, you got two spaces. These things are two homeomorphic and the. De-notation I've been using it's just been the approximately equals to with the little superscript to there are a, there's a close subset of X and a close subset of Y, such that the two close subsets are helping me morphic to each other. And they're open compliments on UX and UI are also me morphic. So the idea is you're generalizing a homeomorphism, things that are homeomorphic are certainly two homeomorphic, but not vice versa. And you aren't just splitting spaces sort of into random pieces and asking about them. That would be too hard to say anything about it. It's just too general. But when you put the extra condition on there, and the split has to occur with one piece open and one piece closed, then you actually do get some nice results out of this. One thing to mention a big chunk of what I've been working on is, is sort of how far apart two homeomorphism and homeomorphism are, as far as relations go. One thing that came up very early on was that homeomorphism is clearly any equivalence relation. If you take a collection of spaces, we don't get the same thing at all for two homeomorphism. In fact, it's got major problems with being transitive. And when I say major, what I mean by that is a lot of the work I've done has been trying to figure out if we can put certain conditions on groups of spaces and actually have them be transitive under two homeomorphism. And that's been pretty hard to actually do these days. There's a lot of failure problems there for even for spaces that are kind of otherwise nicely behaved. Okay, so one way that I took this research was I asked this sort of broad question. If you have some space why it's got property P so P is compactness or connectedness or matrizable or whatever you want. How can we characterize the spaces that are two homeomorphic to that space. And so the first thing I'm going to talk about is the compact spaces. So compactness will be key here. It's worth mentioning just so we get a handle on what this stuff actually means. A really simple example of spaces that are too many morphic but not how many morphic are mega and mega plus one. So a mega mentioned earlier. That's the smallest infinite ordinal. So it's just some sequence without a limit point. If you take mega plus one, you got a sequence with the limit point added onto it. That's certainly compact. But it's not. Oh, mega is not compact. Not homeomorphic, but they are two man morphic. The idea is, if you take that limit point, it's one point in the compact set. So it's a closed set. Toss it over to one point in the original sequence. They're homeomorphic points are homeomorphic just by themselves. What you have left over is just each space now has one copy of a mega left in it. So these are the kind of tricks we do a lot in this in this space. So it's a limit point in a space at large, but taken by itself at just any old point. So compactness is not preserved by too many morphism. My little spoiler alert is here. Not much is most properties aren't actually. So one of my other ways of looking at this is to figure out properties that are preserved. Okay. So I was able to get a compactness result. And it's, it's a nice result in and of itself. It says that a space X is too many morphic to a compact space. If and only if this set that I called S sub X. This is the points where X is not locally compact at these points. If that set itself is compact. So the place where it's not looking back is, it's somehow small. And so that's a, that's a nice result. But let me explain why it can make it even nicer than this. So my, my, my, my, my, my, my thesis is kind of divided into, into two really distinct sections. When I began working, I was working in the area of compacted for. Cations and the stone check back to vacation. And had some results on that for a while. And the quarter got a little bit stymied in our, in our work. And then this paper came along. And it was a nice paper to work off because it had a lot of open questions to it. But I thought it would have been nice to have some sort of connecting thread to kind of bring these two parts of thesis together. So let me tell you what that, what that connecting thread is. So a few notes down here. The reason why we're using tick and off spaces is that these are exactly the spaces that can be embedded into compact house door spaces. So if you have a space X, and you can embed this densely into a compact house or space Y, then we call Y a compacted for. Cation of X. So in the previous example, we have this convergence sequence with a limit point. That's a compactification of a nonconvergent sequence. It takes a sequence adds a point to it. It's dense. And now you've got a compact space. So compactifications exist exactly when you have a tick and off space to start with. And it turns out even better than that. If you take all the compactifications of a certain space, you can order those. And so there's a maximum element, which is called the stone check compactification. So I denote that beta X. That's what I was working on. Sort of in the first half of my thesis. Well, if you take this. Space X and it's non-compact to begin with, then you're going to have to add points to X to get the stone check modification. Okay. So the points you add, they're called the stone check remainder and that's deemed X star. So as long as X is in compact, the stone check remainder has at least one point in it. And so it turns out that this previous theorem, which is, again, it's, it's a nice characterization about not looking compact things. It has an even more attractive version of it. So that's entirely equivalent to the fact that you are too many morphic to a compact space, if and only if your stone check remainder is locally compact. And so I like this result a lot for, for two reasons. The first one is it really helps bridge the two parts of my thesis. So it's almost perfect that it worked out this way. The second thing is dealing with the stone check modification, even of a simple space is usually really hard. I mean, even nice well-behaved spaces have super nasty stone check modifications. And so their remainders are really hard to kind of wait around. Sometimes though, it's actually pretty easy to get too many morphisms. So if it turns out you can get a space to be too many morphic to a compact space, automatically you know something about it stone check remainder. And that's a really powerful result, I think. So I like that result a lot. Let me talk next about scattered spaces. They were a, they were a class of space that I was investigating with respect to this stuff that turned out to be a little bit easier to work with than maybe just general spaces. So I'm going to talk about scattered spaces. Scattered spaces are spaces where every non-empty subset of X contains an isolated point of X. Okay. And in particular, if you've got a scattered space, you can talk about its canter been Dixon height. And so I define what that is down here. It's a, it's a recursive definition. So the canter by Dixon height, you start off with X zero being the empty set. That's not really that interesting, but it's not that interesting. It's not that interesting. It's not that interesting. It's not that interesting. This is scattered. So there definitely points in there. Then X two is the set of non isolated points of X one. So you're throwing away the isolated points of X. And then what's left, what's left over is either now isolated or it's not isolated still. And you keep going up and throwing out the set of isolated points from the previous level until you get to a limit ordinal. And then you define that to be the intersection of all the points. So you define the canter been Dixon height to be the supremum. Where of the muse, where the. Mute level is not empty. So the first level that you get nothing. If you toss everything out, that's what the CB height is. So it can get kind of yucky for large. Cardinal numbers, but turns out here. We only care about things that have finite CB height. So it's not too bad. Um, so. I mentioned earlier that a lot of properties. Aren't preserved by too many morphism. That's true. But one that is, is actually scatteredness. And that's a big help because it lets us look at a, at a class of spaces and then we're sort of stuck with those spaces, which is in, in my view, a good thing has helped restrict down over talking about. So the, the first. And the second. Lama here is if X to my work with the Y and X is scattered, Y is scattered. Um, turns out this is true because being scattered is a rare property that's inherited by both subspaces and finite unions. And so using this little fact, I was able to characterize the things that were. Too many morphic to discreet spaces. It turns out that they're scattered spaces that has to be true by the. Lama. And again, a discreet space has got CB height one. Anything which is too many morphic to a discreet space can't have CB height bigger than two. So at most you can have one level of non isolated points over top the isolated points. And you got to have cardinalities equal to, but that's a requirement for anything to be too many morphic. So that was a nice characterization for discreet spaces. Let me mention a few little examples here. Um, so we already talked about the example of. Omega and. Oh, mega plus one. Um, they're too many morphic ones discreet one isn't so being a discreet space, not preserved. Most things aren't so that's not a big surprise. Um, for a little way to see example. If you take Omega. And rather than plopping on there a limit point at the end, you put on a point in mega star. So that's the stone check remainder of. Omega, right? So lots of points and there's two to the C points in there. So just grab one of them. It turns out you can show that these two spaces are too many morphic. That's easy. Cause points just go to points where one point goes to one point, but also there's a result. Um, if you're a point in the stone check remainder of any space, you are not a point of first countability. And so you see that in the former case. Omega, the weight is a lot of not characters, a lot of not, um, in the latter case, the weight is larger and the character is larger. So you lose weight and character preservation by this example. What I like best about this, though, isn't those facts. Um, I like the fact that what you have here is two spaces that are not homeomorphic. They're too homeomorphic. And they have the same stone check notification. I think that's kind of a neat thing. And that's one question I'm looking at in the future is sort of how far apart spaces can be and still have the same stone check notification under too many morphism. So as an example of one that works. Um, so one question that was asked by my advisor, answered by me was in the case of a discreet space, you can only go up one level. So you can go from CB height one to CB height two. That's the best you can do is that all was the case. In other words, if you're a scattered space and you've got a finite CB height N and you have some other scattered space that's too morphic to it. Is there a finite upper bound? Is it, is it plus one? Or is it N plus one? Is it larger than that? Is there none? Um, turns out that the answer is the finite height exists. It's two N. Um, so, uh, I, I prove that. And then what's interesting there. Um, so you want to have that result that that shows you that there's two finite CB height spaces there at most separated by two N in height if they're too many morphic. Um, but that's not always the case. If you have a space that's got height N, you do not have to have it be too many morphic to a space. It's got height two N. So there are lots of examples where spaces don't achieve the maximal result or even results between there. Um, however, um, what I, what I, what I also showed was that you can construct examples where the maximal heights achieved. So you do in fact, um, get that that is going to be an achievable height. It won't be a strictly less than result. Okay. Um, that's all I have. I don't know whether I've gone over or gone under, but, um, that's it. Thank you. Thanks, Steve. That was the perfect amount of time. Uh, let's, thanks, Steve. It's so awkward. All right. Um, does anyone have any questions? I got a question. Um, so the, the first. Um, okay. I think the first thing you said that the things are too homeomorphic. Um, I think it was like slide six. Yeah. So if, if you know that X star is locally compact, then do you know what the space Y is? No, I mean, um, so at least I don't, I shouldn't say we don't know, but I don't know. Um, so, I mean, the, the, the problem is that, I mean, there's, there's, I mean, lots of things depending on the individual compact space. I mean, in other words, I mean, the best you can say is that they have the same. Um, you can probably say more. I mean, there's, there's probably more you can dig around in there for individual spaces. Like I've played mostly with the, for the classical metric spaces like omega and Q and R and things like that. So I know a little about those, but I mean, for a, for a general space, the answer is, is no, um, and what's even worse than that is, I mean, I, I think the number of spaces where X star is locally compact. I mean, I don't know much about very many, very many stone check remainders at all. And it's, it's generally not known. So I mean, for an arbitrary space, the stone check remainder is, is usually about as mysterious and object as you're going to find. So let's, it's, I would say it's probably easier to go the other route, which is defined to me more person than it is to find out about the remainder. Okay. Thanks. Sure. Any other questions? That's a quick clarification about, I think it was a last slide where you have this bound of, yeah, so in and then less than or equal to two in. So were you, is this a sharp bound where you have spaces with an in and then a Y with a two in? Yeah, absolutely. Yeah. So I mean, you can, so you could even construct spaces where I mean, you can, you can play with things. You can, you can make a space where you'll get every, every level from two and down to be too many morphic to it by pulling off certain things. And there are spaces where you can pull out this and that. So I mean, you can, you can kind of play with things like, I mean, it's, it's relatively easy to construct even small examples where you start with a space that's got a CB height two, and it will not be too much for any space at CB height four. That's not very hard to do just by looking at the one point modifications of discrete spaces, but I mean, beyond that, all the different examples, I mean, they get, they get cumbersome really fast. So my, my patience level war out with those. But it is a sharp bound. Yeah.