With continuity under our belts (albeit with some bumps along the way), we can look at something that many people consider *the* central concept of topology: homeomorphisms.
A homeomorphism is what defines the topological concept of *equivalence*. Remember the clay mug/torus metaphor from from my introduction: in topology, two topological spaces are equivalent if they can be bent, stretched, smushed, twisted, or glued to form the same shape *without* tearing.
The rest is beneath the fold.
*(A very important thing about the intuition above: understanding homeomorphism in terms of deforming topological spaces, isn't really quite correct. The reshaping of topological spaces is more properly modeled as a continuous deformation, which we'll define later in terms of something called **isotopy**.)*
To make it more general, so that we aren't limited to topological spaces that are easily visualized, we can say that two topological spaces are equivalent if they can be reshaped into each other without making any points that are close together in one be separated in the other.
That's a fairly awkward way of saying it. It's actually easier to talk about it
in formal terms - which is where homeomorphisms come into play.
A homeomorphism between two topological spaces **S** and **T** is a function f : **S** → **T**, such that:
1. f is continuous
2. f-1 is continuous
3. f is a bijection (f is total, one-to-one, and onto).
By the continuity of f, we know that the mapping from **S** to **T** is smooth; by the continuity of its inverse, we know that the mapping from **T** to **S** is smooth; and since it's a bijection, we know that every point in **S** is mapped to exactly one point in **T**, and every point in **T** is mapped to by exactly one point in **S**.
A homeomorphism is also known as a *topological isomorphism*: it's an isomorphism between two topological spaces - that is, a function between two topological spaces that preserves all of their topological properties. Every homeomorphism is also an iso-arrow in the category of topologies: everything we know about iso-arrows in categories can be specialized to apply to the relationships between topological spaces defined by homeomorphisms.
There are some things about homeomorphisms that can seem a little counterintuitive. Homeomorphisms determine a kind of topological equivalence: they preserve all topological properties. But we often associate properties with topological spaces that are *not* really topological. We often think of topological spaces in terms of metric spaces that have topological properties. But *metric* properties are *not* topological properties, and so a homeomorphism can modify the metric properties of a space.
Homeomorphisms are fascinating things. They give you an ability to grasp what topological properties really mean in a fascinating way. A few examples:
* A sphere and a cube are the topologically identical. Not too surprising, but it sometimes throws people that a hard angles on a cube can turn into a continuous curves of a sphere.
* A more interesting one: you can "shrink" the infinite to a finite range! For any number *n*, the open numeric interval from -n to n is homeomorphic to the set of all real numbers.
* And finally, a really fascinating one that *I* at least was totally surprised by when I first learned it: take a sphere. Remove *one* point. What you get is homeomorphic to a plane. (I just find this one too cool for words: in topology, a sphere is only different from a plane because of *one* point!)
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It's the point at infinity that gets removed, isn't it? That is to say, paths that go off to infinity on the plane are mapped to paths approach the "missing" point on the sphere, right?
"A more interesting one: you can "shrink" the infinite to a finite range! For any number n, the open numeric interval from -n to n is homeomorphic to the set of all real numbers."
Shucks, RF electrical engineers do this all the time. This is the exact point of the Smith Chart:
http://en.wikipedia.org/wiki/Smith_chart
See, I think that a neater homeomorphism is the one between a wrap-around computer screen and a torus, but I admit that's a matter of taste.
(What do I mean by a "wrap-around computer screen"? I mean something like you had in the old asteroids game, where anything that slides off to the left appears in the same spot on the right, and vice-versa, and the same for the top and bottom edges)
Canuckistani:
Yes, that's basically it. It's just another of the "shrinking infinity to a finite range" things: the flat plane is infinite; the sphere with a point removed is finite. But you can reduce the plane to the almost-sphere, or flatten and stretch the almost-sphere into the plane.
Daniel:
The computer screen/torus one never excited me; I think because the way that a torus was first explained to me was "take a piece of paper, connect the top to the bottom and the left to the right". And since I was a precocious little bugger, and didn't get my first computer until my senior year of high school, when I first saw games that did the wrap-around thing, my reaction was "Oh, it's played on a torus." So I tend to think of it as a sort of natural/obvious one. But my mind is warped, so what's obvious to me might be amazing to others; and some of the stuff that amazes me seems downright pathetic to other people :-).
(To me, the ability to read a map is amazing. I can't do it, and it never ceases to amaze me to watch people extract so much useful information from something that to me is just a bunch of wigglies.)
I'm with Mark on the torus one, although it reminds me of an interesting puzzle: did anyone ever play the Get Red Spheres minigame in Sonic 3 and Sonic and Knuckles? That was played on a most odd surface - it curves away like a sphere in all directions, but wraps like a torus (which it can't be because a torus would be seen in the distance, as evidenced by the Halo games). Any thoughts?
Given that there exist non-surjective epimorphisms in Top, such as the inclusion of Q into R, can you give a characterization of epimorphisms in the category?
a sphere is only different from a plane because of one point!
Ooh! Ooh! Do an essay on projective geometry!
Fractals, here we come :)
Well, the homeomorphism between the plane and a sphere with one point missing confirms something that I said in one of the older topology posts; at least, I think it does. It was about what would happen if you took a sphere and simply poked a hole in the surface, without going all the way through with a tube. I said that it's not the same as a torus - that at the edge, there is no longer a proper neighborhood of points. I said that it was no longer a 2-manifold, but it appears I was wrong. It is, but it's equivalent to an infinite one, which still makes it distinct from a torus or sphere.
Well, if you remove a closed set, then you still have a manifold, pretty much by the definition of open sets. Generally, the most interesting manifolds seem to be the compact ones, like a sphere and a torus but not a plane. But yeah, planes, cylinders, and the likes are manifolds just the same.
A little thread necromancy here. I just noticed Alon's comment and wanted to reply.
Actually, the inclusion i of Q into R is not epi in Top. Let X be the two point indiscrete space (that is, X contains two points "a" and "b", and the open sets are {} and {a, b}). The nice thing about an indiscrete space is that every function into it is continuous. So let f be the function from R to X that maps every point to a, and let g be the function from R to X that maps the rationals to a and the irrationals to b. Then f(i(x)) = g(i(x)) = a for all x in Q, but f != g.
This counterexample relies on the fact that X is not Hausdorff. If you restrict your category to Hausdorff topological spaces, then it's not hard to show that a map is epi precisely if its image is dense.