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How important is topology for modern mathematics? 
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#1
Aug911, 06:54 PM

P: 334

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? 


#2
Aug911, 07:17 PM

Sci Advisor
P: 1,807

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


#3
Aug911, 08:23 PM

P: 491

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



#4
Aug911, 08:31 PM

P: 334

How important is topology for modern mathematics?



#5
Aug911, 08:36 PM

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PF Gold
P: 9,415

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.



#6
Aug911, 08:36 PM

P: 491

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.



#7
Aug911, 08:48 PM

P: 491




#8
Aug911, 09:26 PM

Mentor
P: 18,330

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


#9
Aug911, 09:30 PM

Sci Advisor
P: 1,807

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



#10
Aug1011, 08:59 AM

Emeritus
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PF Gold
P: 9,415

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). 


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