Intuition on "giving a set a topology".

[center o bass]

The intuitive picture I have of giving a set a topology, is that of giving it a shape in the sense of connecting the points and determining what points lie next to each other (continuity), the numbers of holes of the shape, and what parts of it are connected to what.

However, the most abstract notion of giving a set ##X## a topology, is that of determining what subsets of ##X## are open subsets. Can I get these two to correspond to each other?

I.e. explain in what sense determining open subsets correspond to giving a set of points it's shape as described above?

If not in the most general case, is this possible for topological manifold (that looks like R^n)?

 

[gopher_p]

I think your intuitive understanding might need some refinement. Or maybe the language that you've chosen isn't as expressive of your intuition as you need it to be to communicate it unambiguously.

Any topology that distinguishes points distinguishes them equally, except for perhaps being able to tell which points are in the same/different connected components. For instance the standard topology on ##\mathbb{R}## can't tell that ##0## is closer to ##1## than it is to ##100##. So the topology only tells you which points are in the same galaxy (connected component). It doesn't tell you how close the points are or how close the connected components are. You need a metric for that kind of information.

Also a topology by itself doesn't tell you whether there are "holes"; i.e. you can't tell just by looking at the open sets in a topological space where the holes are. Think about the topology on the unit circle inherited from the standard topology on ##\mathbb{R}^2##. Or look at open sets ##\mathbb{R}^2\setminus\{0\}## with the "same" topology (i.e. the one induced by the standard metric restricted to ##\mathbb{R}^2\setminus\{0\}##).

And while it's technically true that a topology tells you something about connectedness, that's only because we've defined connectedness in terms of open sets. But that defintion turns out to be so far from the "intended" meaning of "connected" that we had to go and make up this thing called "path connectedness" to more accurately reflect our intuition of what it "should" mean for a set to be connected. Of course this intuitive version of connectedness can't be determined solely by looking at the open sets.

As for a topology giving a set its "shape" I'll just refer you to the Generalized Poincare Conjecture (http://en.wikipedia.org/wiki/Generalized_Poincaré_conjecture) and tell you that the intepretation for n=2 is that the surface of every object that you have ever seen that doesn't have a hole is the same topologically. By "has a hole" I mean that you could run an elastic string through the object and tie the ends together and not be able to separate the string from the object no matter how much you stretched the string. Or to put it another (extraordinarily ridiculous) way, if you gave everything a body suit (to cover up the holes) all of the shapes have the same surface topologically speaking. So if you had the right kind of material, one that was sufficiently elastic and malleable, you could make one pattern of "body suit" that would work, without tearing or folding or resewing, for every conceivable 3-d object that exists or could exist in ours or any universe. If you still think that a topology really gives you any information about a set's shape, then I'd have to say you have a very relaxed definition of shape.

 

[homeomorphic]

Unfortunately, I have to disagree somewhat.

Any topology that distinguishes points distinguishes them equally, except for perhaps being able to tell which points are in the same/different connected components. For instance the standard topology on R can't tell that 0 is closer to 1 than it is to 100. So the topology only tells you which points are in the same galaxy (connected component). It doesn't tell you how close the points are or how close the connected components are. You need a metric for that kind of information.

True so far.

Also a topology by itself doesn't tell you whether there are "holes"; i.e. you can't tell just by looking at the open sets in a topological space where the holes are. Think about the topology on the unit circle inherited from the standard topology on R2. Or look at open sets R2∖{0} with the "same" topology (i.e. the one induced by the standard metric restricted to R2∖{0}).

If you are more precise about what you mean by "holes", then yes, they are determined by the topology, given a suitable definition (open sets). In fact, it's even better (worse?) than that. The holes are determined by the homotopy type, which is even LESS than the topology. In particular, I'm referring here to the kinds of holes that are measured by algebraic topology invariants, like homotopy groups and homology groups. I think that is what center o base had in mind.

And while it's technically true that a topology tells you something about connectedness, that's only because we've defined connectedness in terms of open sets. But that defintion turns out to be so far from the "intended" meaning of "connected" that we had to go and make up this thing called "path connectedness" to more accurately reflect our intuition of what it "should" mean for a set to be connected. Of course this intuitive version of connectedness can't be determined solely by looking at the open sets.

I don't think I quite agree that the definition of connnectedness is so removed from intuition. First of all, if you have local path-connectedness, then path connectedness is equivalent to connectedness. Secondly, if you have an intuitive idea of what an open set ought to be, it makes sense that open sets can separate things. In particular, if you are dealing with a special case like subsets of R^n, it makes perfect sense in most cases, although there are some nasty counter-examples like the topologist's sine curve that show that there is some subtlety to it. It's true from a certain point of view that, in the general case, open sets have to do with connectness sort of only because we defined them that way. However, that doesn't imply that it's not intuitive. It's actually an implementation of the intuitive idea. In particular, when I think about point-set topology, I usually draw pictures of everything. The pictures are actually an essential part of my ability to write the proofs. So, the intuition ultimately DOES lead to correct reasoning about the possibly arbitrarily defined topology, even if that intuition still needs to be checked and confirmed by proof.

As for a topology giving a set its "shape" I'll just refer you to the Generalized Poincare Conjecture (http://en.wikipedia.org/wiki/General...%A9_conjecture ) and tell you that the intepretation for n=2 is that the surface of every object that you have ever seen that doesn't have a hole is the same topologically. By "has a hole" I mean that you could run an elastic string through the object and tie the ends together and not be able to separate the string from the object no matter how much you stretched the string. Or to put it another (extraordinarily ridiculous) way, if you gave everything a body suit (to cover up the holes) all of the shapes have the same surface topologically speaking. So if you had the right kind of material, one that was sufficiently elastic and malleable, you could make one pattern of "body suit" that would work, without tearing or folding or resewing, for every conceivable 3-d object that exists or could exist in ours or any universe. If you still think that a topology really gives you any information about a set's shape, then I'd have to say you have a very relaxed definition of shape.

That's true, but on the other hand...

http://en.wikipedia.org/wiki/Mostow_rigidity_theorem

So, you see, under certain (admittedly, rather restrictive) assumptions, topology can, together with other things, indeed, have quite a bit to say about the shape. Everything, in fact.

I do agree, though, that I don't like the word "shape" being associated with topology because shape tends to be associated with geometry/distances/proportions, none of which is present explicitly in the topology itself.

To answer center o base, when you specify a topology you specify which sets are open. That much you've said. I'd interpret the general definition of topology as a generalization of cases like R^n and that of surfaces where we already know what topology is like. In particular, for spaces like R^n or a surface, we don't even need the most general definition of topology to study their topology (although it is handy for dealing with constructions like quotient spaces). In R^n, we already have a perfectly good notion of continuity in terms of the epsilon-delta definition. So, we already know, in that case, what a homeomorphism means without knowing the general definition of a topology. A homeomorphism is simply a continuous map that has a continuous inverse.

BUT WAIT, THERE'S MORE...

Not only do we know what homeomorphisms are, we also know what open sets are. An open set like a generalization of an open interval. Roughly speaking, it doesn't contain boundary points, and more precisely, a set is open if every point in the set has some epsilon-ball around it that lies completely in the set. And with this definition of open set, it's a theorem that a map is continuous if and only if the pre-image of an open set is open. And THAT gives us a way to extend the notion of continuity to less familiar settings (function-spaces, moduli-spaces, quotient spaces, etc). We just say which sets are open and that defines which functions are continuous.

So, that's where the answer really lies. Continuity in the laymen's sense (well, the formalized, real analysis version of it, anyway) is equivalent to pre-images of open sets being open. So, look up that proof or derive it for yourself again and think really hard about it and then you'll understand what the connection is between your sort of "shape" and the open sets.

Think about the example of surfaces. Two surfaces are topologically equivalent if you can deform one continuously into another (the fine print here is you are allowed to cut, as long as you glue back together in the same way as it started). More precisely, if there is a homeomorphism between them. In other words, a continuous map with continuous inverse. If they are embedded surfaces, you can just use epsilon-delta continuity to make sense of it. So, basically, a map is continuous if y only moves a little bit if x only moves a little bit. No jumps. It just so happens that that is equivalent to the pre-image of open sets definition in the case where the epsilon-delta one makes sense.

 

[gopher_p]

With regards to "holes", my point was that it requires extra information beyond the topology. Determining the homotopy type requires information from "outside" of the topologcal space. I could be wrong, but don't you need to jump to a completely different category to even start talking about homotopy?

As far as connectedness goes, I was thinking more along the lines of the rational numbers with the topology inherited from the reals. The intuition (at least in my mind) is that the set is totally disconnected when the topological reality is exactly the opposite.

I suppose I was coming from a point of view that the topology was just a jumping-off point for talking about these more geometric ideas. The topology itself doesn't tell us much about what a set "looks like", but it does give us the base information that we need in order to begin to talk about which functions are paths, which ones are appropriate for establishing homotopy type, which ones might constitute charts (in the case where we have a manifold), which ones are homeomorphisms, etc. I suppose it's just a difference in philosophical perspective rather than mathematical.

But you're the topologist, and will defer to your perspective.

 

[homeomorphic]

With regards to "holes", my point was that it requires extra information beyond the topology. Determining the homotopy type requires information from "outside" of the topologcal space. I could be wrong, but don't you need to jump to a completely different category to even start talking about homotopy?

Well, not really. If two spaces are homeomorphic, that implies they have the same homotopy type. That's a different category, but it's just a category in which more things are equivalent. But if they were already equivalent in the topological category, then they are still homotopy equivalent. The homology and homotopy groups ARE topological invariants. When you say they are homotopy invariant as well, that's a STRONGER statement than being topologically invariant. You don't need to know about the category of homotopy types to define homology or homotopy groups. That's all just defined in terms of continuous functions.

As far as connectedness goes, I was thinking more along the lines of the rational numbers with the topology inherited from the reals. The intuition (at least in my mind) is that the set is totally disconnected when the topological reality is exactly the opposite.

I did actually have this in mind, too, when I said there are some subtleties. But on closer inspection, I don't think it's that surprising. It is disconnected, just not totally, so it's not the opposite. Take any irrational number. Then all the rationals below that guy and all the rationals above him form a separation. So, it's not connected. I think my intuition is happy with that. But you're right, it's not totally disconnected. Any open set has to contain lots and lots of rationals, not just one, so each rational, by itself can't possibly be open. On closer inspection, not so unintuitive. It just might not be your initial, gut reaction. I guess it's a little weird that you can still separate things like I said, as close as you want to a given rational. But you just have to incorporate counter-examples like this into your intuition to get the full picture. On the other hand, maybe from another point of view, your gut-reaction SHOULD be that it's going to be something in between connected and totally disconnected.

I suppose I was coming from a point of view that the topology was just a jumping-off point for talking about these more geometric ideas. The topology itself doesn't tell us much about what a set "looks like", but it does give us the base information that we need in order to begin to talk about which functions are paths, which ones are appropriate for establishing homotopy type, which ones might constitute charts (in the case where we have a manifold), which ones are homeomorphisms, etc. I suppose it's just a difference in philosophical perspective rather than mathematical.

Well, topology is extra structure on a set. So, in a way, it DEFINES some very limited aspects of what sets look like, but it's kind of analogous to the fact that a figure may be made from clay. The clay is the set. The topology is what it becomes, after the sculptor has done his work. But of course, the sculpture is only considered up to homeomorphism, so we don't actually have much information about what it looks like, but we have a little. But the clay (the set) by itself doesn't really "look" like anything in the first place because it's got even less structure than a topological space.

 

[WWGD]

Actually, from the research of Dan Quillen , you can talk about homotopy and homotopy equivalence in a "reasonable sense" in any model category http://en.wikipedia.org/wiki/Model_category , and not just in top, or one of chain complexes.

 

[homeomorphic]

Yes, I'm vaguely familiar with model categories from my algebraic topology friends' talks. I suppose that's an interest aside, but my point was just that topological spaces are definitely enough to define homotopy and homotopy equivalence. No need for more. There is more out there, but you don't need it if you are just interested in topological spaces.

But, yeah, if you want to generalize homotopy to other categories, you can.

 

[WWGD]

Yes, sorry, I just thought I would put it out there; I benefit from others' similar asides.

I found it interesting to work with initial and final topologies for collections of functions, and figuring
out "extremal" topologies re some property, i.e., say we have a chain ( by inclusion ) of topologies
on a given space X : is there a maximal topology ( in some chain) for which X is compact, connected, complete,
etc. Tho maybe this is more of interest to functional analysts or people who do point set topology ( I wonder
if anyone still does, at least at a research level ). If you want to sharpen your point-set- theoretical skills, try
as an exercise to find the maximal and minimal non-trivial ( i.e., neither discrete nor indiscrete ) topologies
on a given space.