2
votes
1answer
39 views

Why do we require $X$ to be Hausdorff when defining $C(X)$?

In Efton Park's "Complex Topological K-theory", he begins a section by defining the Banach algebra $C(X)$ (continuous functions $f: X \rightarrow \Bbb C$), presupposing that $X$ is Hausdorff and ...
6
votes
4answers
705 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
0
votes
0answers
29 views

What kind of norm is it in the definition of $S^{n-1}$?

Definition (wikipedia) $S^n\triangleq\{x\in\mathbb{R}^{n+1}: ||x||=1\}$ is said to be a 'n-sphere' What norm is it referring to ? I have proven that ...
7
votes
1answer
256 views

What is the significance of limit points?

When I had my first taste of topology a couple of years ago, our lecturer emphasized the following notions. closed set, closure, closure point open set, interior, interior point Of course, these ...
5
votes
4answers
298 views

Why the axioms for a topological space are those axioms?

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard ...
0
votes
2answers
153 views

motivation of limit points

Lets use Wikipedia's definition of a limit point and let $\lim(A)$ denote the set of limit points of $A$. $a\in \lim (A) \leftrightarrow a\in\operatorname{cl}(A\setminus\{a\})$, $\lim (A)\cup A = ...
22
votes
4answers
1k views

Motivation behind topology

What is the motivation behind topology? For instance, in real analysis, we are interested in rigorously studying about limits so that we can use them appropriately. Similarly, in number theory, we ...
4
votes
3answers
285 views

Motivation for the Mapping Class Group

Question: What is the motivation for studying the mapping class group? In particular, what types of questions does it attempt to answer and what kind of invariant is it? Motivation for this ...
39
votes
7answers
3k views

What's the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?

Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) ...