# Tagged Questions

The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness.

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### How to define metric in the Space of Holomorphic Functions?

I am looking for a proper way to define distane on the space of Holomorphic functions defined on a domain $D$.Does the Montel's Theorem (Given below from Stein's Book) helps to Characterize Compact ...
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### Why is $\Bbb R\setminus\{\frac1n\mid n\in\Bbb N\}$ not locally compact?

I have a question: if I take in $(\mathbb{R},|.|)$ the set $A=\left\{\frac1n, n\in \mathbb{N}\right\}$ and I consider the set $B=\mathbb{R}\setminus A$ I want to prove that $B$ is not locally ...
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### Existence of a open set between a compact and an open set

Let $M$ be a compact manifold, $K\subset M$ compact, $U\subset M$ open. Does in this case always exist a open set $V\subset M$ such that $K\subset V\subset\bar{V}\subset U$ ?
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### Compacts And The Reciprocal Of The Weierstrass Theorem

While I was studying Functional Analysis, this question arised: Let $K \subseteq \mathbb{R}$ be a subset with the propertie that, for all $f$ continuous ($f \in \mathcal{C}^0(\mathbb{R},\mathbb{R})$...
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### example of a particular topological group

Can someone give an example of a topological group $G$ that is not Hausdorff but that contains a fundamental system of neighbourhoods of $1\in G$ consisting of quasi-compact subgroups? Thanks in ...
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### Point-wise bounded and equicontinuous sequence of functions has a uniformly convergent subsequence

Problem We have a sequence $(f_n)$ of continuous functions on a compact metric space K. It is also given that $(f_n)$ is point-wise bounded and equicontinuous. Now show that $(f_n)$ has a sub-...
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### Two (maybe nonequivalent) definitions of local compactness

$X$ is locally compact if every point has a neighborhood with a compact closure. $X$ is locally compact if every point lies in the interior of a compact subspace of $X$. Clearly, $(1) \implies (2)$. ...
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### Is every compact totally ordered space homeomorphic to a subset of $[0,1]$?

Let $(X,\leq)$ be a totally ordered set such that, equipped with the order topology, $X$ is compact. Is then $X$ homeomorphic to a closed subset $A \subseteq [0,1]$? A way to ask this question ...
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### Space on which all real-valued continuous functions achieve maximum but not compact?

A friend is writing a book for non-mathematicians; he has asked me some questions... One possible direction I suggested was whether a topological space (metric space can probably be assumed given what ...
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### Why do we need tube lemma to prove the compactness of the product of two compact spaces?

I read the proof in Munkres' book Topology which uses the tube lemma but still thinking about an easier proof using basis of product topology : $X \times Y$ has \{B_x \times B_y, B_x \times Y, X \...
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### Show that representative functions on a profinite group factors. [duplicate]

Let $G$ be a compact group. A representative function $f\in\mathcal{C}(G,\mathbb{K})$ is a function such that $\dim\left(\operatorname{span}\left(Gf\right)\right)< \infty$. Remark that the ...
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### Closed and boundary subsets

Let $X$ be a nonempty compact space and let $F_1, F_2, ...$be its closed and boundary subsets. Prove that $\bigcup_{n=1}^{\infty} F_n \neq X$ I have no idea how to do it. My only plan would be to ...
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### Countable and not closed subset of infinite compact space

The taks is: Show that in every infinite compact space there is a countable subset that is not closed. At first I read that it should be closed and I had an idea to take a point $x_1 \in X$ and an ...
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### Is the intersection of two locally compact subspaces locally compact?

Taking locally compact as such that every point has a local base of compact neighborhoods, is the intersection of two locally compact subspaces locally compact?