5
votes
2answers
100 views

Ways to induce a topology on power set?

In this question, two potential topologies were proposed for the power set of a set $X$ with a topology $\mathcal T$: one comprised of all sets of subsets of $X$ whose union was $\mathcal T$-open, one ...
3
votes
1answer
109 views

A question about the hierarchy of topologies on a given set

It's easier to understand with examples: Every finer topology than a Hausdorff topology is hausdorff. Every coarser topology than a compact topology is compact. What are the full set of properties ...
14
votes
4answers
737 views

Can we get un-obvious results by defining sophisticated topologies?

What I originally found so interesting about general topology was how general a type of thing a topology is, and how the terminology open, closed, compact, continuous, convergence et cetera means ...
1
vote
3answers
140 views

Uniquely geodesic spaces

The purpose of this list issue is to better understand the class of uniquely geodesic spaces. I'm looking for two different things : Overclass : for example geodesic space or contractible space. ...
2
votes
1answer
107 views

Tychonoff Theorem and the axiom of choice

How to show that The Tychonoff Theorem and the axiom of choice are equivalent? Here I want to collect ways to prove it. Thanks for your help.
0
votes
1answer
250 views

If $X$ is complete and totally bounded, then $X$ is compact [closed]

Let $X$ be a metric space. Whar is your favorite way to show: If $X$ is complete and totally bounded, then $X$ is compact? Thanks for your help.
16
votes
11answers
3k views

How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
2
votes
0answers
51 views

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable?

Are there many spaces which have a regular $G_\delta$-diagonal but is not submetrizable? Submetrizable = if we can choose a coarser topology on the space $X$ and thus make it a metrizable space. ...
6
votes
2answers
86 views

Collecting definitions of continuity.

Let $X$ and $Y$ denote topological spaces and consider a function $f : X \rightarrow Y$. I'm collecting possible definitions/characterizations of the statement "$f$ is continuous." Here's two to get ...
6
votes
3answers
259 views

Why are ordered spaces normal? [collecting proofs]

Greets This is a problem I wanted to solve for a long time, and finally did some days ago. So I want to ask people here at MSE to show as many different answers to this problem as possible. I will ...
18
votes
7answers
4k views

Any open subset of $\Bbb R$ is a countable union of disjoint open intervals. [Collecting Proofs]

This question has probably been asked. However, I am not interested in just getting the answer to it. Rather, I am interested in collecting as many different proofs of it which are as diverse as ...
2
votes
2answers
147 views

How many ways to construct a dense subspace?

For any topological space $X$, as the title explains, how many ways to construct a dense subspace of $X$? For example, we can construct a dense subspace which is the union of disjoint open subsets of ...
0
votes
1answer
106 views

a question on countable discrete set

Let $X$ be Hausdorff and $C$ is countable discrete in $X$ and $x \in cl(C)$. Does there exist a subset $D$ of $C$ such that $x \in cl(D)$ and there is a point-finite family $\{U_n\}$of open sets ...
9
votes
2answers
263 views

What are some motivating examples of exotic metrizable spaces

Among topological spaces, the metric spaces are usually considered to be the tame animals. Describing the topological notion of closeness by a distance is so intuitive (as opposed to the abstract ...
8
votes
4answers
854 views

Examples of fundamental groups

I'm starting to study fundamental groups and I didn't find in the books of Algebraic Topology many examples of them. Can you list the examples you know and the demonstrations? I think it would be ...
1
vote
0answers
66 views

Partition of open sets in $\mathbb{R}^d$.

Let $\Omega\subset\mathbb{R}^d$ be open. We want to find a good partition of $\Omega$ into more elementary sets. In particular we want compact sets $K_j$'s and open sets $V_j$'s such that ...
63
votes
18answers
6k views

Your favourite application of the Baire Category Theorem

I think I remember reading somewhere that the Baire Category Theorem is supposedly quite powerful. Whether that is true or not, it's my favourite theorem (so far) and I'd love to see some applications ...
12
votes
9answers
928 views

Reference for general-topology

Though there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or ...
1
vote
1answer
321 views

Properties of generalized limits aka nets

I want to find some article or a book which contains all general properties of nets. Of course some of them similar to properties of sequences with almost the same proofs, but I don't fill the edge, ...
32
votes
12answers
10k views

Real life applications of Topology

The other day I and my friend were having an argument. He was saying that there is no real life application of Topology at all whatsoever. I want to disprove him, so posting the question here What ...
2
votes
2answers
108 views

Topologies on spaces of mappings

Given two topological spaces $X, Y$, the only example I know of a topology on the space $\mathcal C(X,Y)$ of continuous mappings from $X$ to $Y$ is the compact-open topology. However I presume that ...