How important is topology for modern mathematics?

kramer733

And what's considered modern mathematics? I always thought it was 1960s+. Around 50 years ago till now is what i considered modern math.

Anyway, how important is topology? I've heard people say "the idea of evolution to biology is the same as the ideas of topology to mathematics." So is it really that important?

disregardthat

well...

yes

You need solid knowledge of topology to study any advanced field of mathematics.

zhentil

Evolution is the unifying principle in biology. Topology is the third most pervasive branch of mathematics.

kramer733

What are the other ones?

Fredrik

A lot of theorems in real analysis are just corollaries of theorems in topology. Most books on functional analysis are impossible to even begin to read if you don't know topology really well.

zhentil

Analysis and algebra, where the ordering is a matter of taste. As a geometer, I say analysis, but I'm sure 50+-epsilon% of mathematicians disagree with me.

zhentil

This seems a bit backward. Do you mean that the first half of a point set topology book is a generalization of the concepts and results of metric spaces? Also, I didn't need more than the definition of a quotient space when I took functional analysis. Topology to me means homotopy and cohomology. I doubt it could be argued that point set topology has had a profound impact on modern mathematics.

micromass

Uuh, you mustn't have gone far in functional analysis then... Functional analysis requires quite a lot of point-set topology. Separability, compactness, Tychonoff theorem, Urysohn lemma, etc., you really think you don't need this?

In practically every branch of math that I know, I use point-set topology: geometry or analysis. Topology had a huge impact on math.

disregardthat

This is not a discussion of chronology or impact, it's about the actual usage in mathematics. Point set topology is a must.

Fredrik

I if you meant to metric spaces, then yes. For example, the result "a subset of the real numbers is compact if and only if it's closed and bounded" can be thought of as a corollary of the slightly more general "a subset of a metric space is compact if and only if it's complete and totally bounded".

As micromass said, it doesn't take long until you need some of the deep theorems like Urysohn, Tychonoff, etc., but what I really meant was that if you can't quickly prove e.g. that a subset of a metric space has compact closure if and only if it's totally bounded, or that a compact Hausdorff space is normal, don't even bother opening Conway's book.

They're not all as difficult to read as Conway, but you certainly need a lot of theorems about metric and topological spaces no matter what book you choose.

(Apologies if I got any of those theorems wrong. I don't feel like thinking it through right now).

OK, that's not what it means to me.