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 definition 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.