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Ok... So I went to a math lecture today and they were talking about topology... I caught bits of it, but because they were trying to prove something very specific, they skipped over some of the basics... The one thing I really wanted to gain from this lecture seemed to be skipped over completely. That question is what exactly is topology? I assume it has something to do with space from the things I have seen... But I am curious to find out more. Any information would be usefull and quite welcomed. 3nodding Thank you for your time.

That band around the donut and the sphere was the lecture. (I think the lecture was titled, why a sphere is not homeomorphic to a torus... But I am not totally sure if that was the name...) 3nodding The other terms are familiar...

For a blurb definition of topology, point-set topology in particular, probably the best one is "the study of properties preserved under continuous mappings." Pretty much every topic in topology is some sort of extension of this--e.g., in considering homotopies, one takes continuous deformations of curves.

First, recall the calculus definition of continuity at x: a function f is continuous at x iff whenever specifying "how close" you want to be to f(x), you can find an interval around x such that all y in that interval are no farther from f(x). Formally, (Vε>0)(Eδ>0)(|x-y|<δ → |f(x)-f(y)|<ε), where I have used V for "for all", E for "there exists, and → for "implies". Just to reiterate,,for every ε-neighborhood of f(x), there is a corresponding δ-neighborhood that whose image is in the ε-neighborhood of f(x).

Now, abstract away from Euclidean space and consider a general metric spaces (X,d) and (Y,ρ), with metric d and ρ, respectively, with some function f from X to Y. Instead of intervals around x or f(x), you have an "open ball" B(x;δ) = {y in X: d(x,y)<δ} of all points within a distance δ of x. Your continuity condition now looks like this: (Vε>0)(Eδ>0)[f(B(x;δ))⊂B(f(x);ε)], i.e., all points within the δ-ball around x map into the ε-ball around f(x). For a metric space, the induced topology is defined as the collection of all sets that are unions of open balls; these sets are called "open sets." The empty set φ is vacuously open (being the union of an empty collection), and the entire space is open (since B(x;1) is an open ball and the whole space is the union of all such balls). In R, (under the standard metric d(x,y) = |x-y|), this means that every open interval is open, but intervals like [0,1) are not. Let's take a look at continuity in a bit more detail if the f is continuous everywhere:
1. The preimage of a f(x), i.e., all values that map to it, includes x and possibly some other points x_α all having f(x_α) = f(x). But since f is continuous everywhere, we can find open balls around each of them that map to B(f(x);ε) for any given ε>0. Take their union--the preimage of an open ball is an open set.
2. Since every open set in Y is a union of open balls, and by (1), their preimages are open sets, take the union again--the preimage of an open set is an open set.
3. Aha! Globally continuous functions are precisely those for which the preimage of an open set (in Y) is another open set (in X).

Even futher, get rid of the metric altogether and consider the structure of the open sets themselves in some space X. They have three main properties: (1) φ and X are open, (2) the union of any collection of open sets is open, and (3) the intersection of any finite collection of open sets is open. Now, call the collection of these sets a topology, and keep the definition of continuity (3), now applied in a more general context with no mention of any metric whatsoever.

Ex.: Is there a topological space X such that every function f from X to Y is continuous, for every topological space Y?
Ex.: Are there topologies that are not induced by any metric?

Mrph. That doesn't mean I have to like it.



Really? Hmm. I guess it might have been my background, but the objects you find in point-set are usually more bizarre, from my experience. I mean, with diff top you at least are confined to manifolds, and alg. top, at least the concept behind it, was fairly intuitive to everyone I know. The idea of spaces that aren't Hausdorff, or sequences that converge to everything, or the non-equivalence of the various topologies on infinite products, always seems to throw people.

Am I allowed to guess? I have an idea, but not a proof just yet (running out of time before I need to do other things).

The discrete metric 'cos every open neighborhood around a point contains solely that point (when the neighborhood is taken small enough) and so

f(B_d(x)) = f(x) which is contained inside B_e(f(x)) for all e

So every map from the discrete metric is continuous.