# Tagged Questions

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### Properties of reflexive Banach spaces

I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily ...
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### Important applications of the Uniform Boundedness Principle

There's like three applications of the uniform boundedness principle in wikipedia: 1) If a sequence of bounded operators converges pointwise to an operator, then the limit operator is also bounded, ...
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### Good topologies on $\mathcal{P}(X)$

Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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.
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### 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.
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### 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 $[a,b]$ ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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 ...