# 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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### Is Inverse of a function continuous too?

I read an example from "Principles of Mathematical Analysis" by Rudin under the section 'Continuity and Compactness'. According to the example, Let $X$ be the half-open interval $[0,2\pi)$ on the ...
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### Open interval $(0,1)$ is totally bounded

Is true that an open interval $(0,1)\subseteq \mathbb{R}$ totally bounded? I think it is not true. Since there is an homeomorphism from $(0,1)$ to $\mathbb{R}$ and $\mathbb{R}$ is not totally ...
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### What is a finite subcover of $[0,1]^{[0,1]}$?

According to Tychonoff's theorem, under the standard topology, $[0,1]^{[0,1]}$ is compact. However, I cannot think of a finite subcover of this space. Also, how does this reconcile with the fact that ...
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### Compact subsets of a Hausdorff space

Reviewing for qual: Let $X$ be a Hausdorff space, $K$ a nonempty compact subset of $X$, and $x \in X\backslash K$. Prove that there exist disjoint, open subsets $U$ and $V$ such that $K \subset V$...
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### Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism [closed]

Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism Requirements for a homeomorphism $f:X \rightarrow Y$: $f$ is ...
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### $K$ compact metric space, is there a finite set of continuous functions that separates points in $K$?

Definition: A family of functions $\mathcal{F}$ on a set $X$ separates points in $X$ if for every distinct pair $x,y\in X$ there exists $f\in\mathcal{F}$ such that $f(x)\neq f(y)$. Let $K$ be a ...
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### Question on proof of unit ball in $C([0, 1])$ not being compact

Take the sequence $f_n(t)=t^n$, $0\le t\le 1$. Then $\{f_n\} \subset \overline{B(0,1)}$, but we have no subsequence of $\{f_n\}$ converging in $C([0,1])$. So the unit ball is not compact in $C([0,1])$?...
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### Non-compact subsets of a metric space $(X,d)$.

I'm trying to come up with an example of a metric space $(X,d)$ such that a subset $A \subset X$ is not compact, but is closed and bounded. Essentially I want to find an example that shows that a ...
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### How does the compactness property help us show a subset $A$ of a metric space $X$ is closed?

We have a compact subset $A$ of a metric space $X$ and we want to show that this implies that $A$ is closed. Let $y \in A$ and $y \in A^c$. For each $y \in A$, we can take open neighbourhoods $U_y$ ...
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### Regularly open, co-zero sets in compact Hausdorff spaces

It follows from the definition of a completely regular space that such spaces have a base consisting of co-zero sets, that is, sets whose complement is the zero set of some real-valued, continuous ...
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### Showing a metric space is not complete.

Consider the metric space $$B = \{ f \in C[0,1] : \int_a^b \left| f(x) \right| dx \leq 1\},$$ where $d(f,g) = \int_0^1 \left| f(x) - g(x) \right|dx$. I'm trying to show that this metric space is not ...
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### Hausdorff compact problem

Let $X$ a Tychonoff space and the topological immersion $e: X \to \prod_{s \in S} [0,1]$. For this other question: Show that for all compact $K$ and for all continuous function $f:X \to K$, there is ...
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### Characterizing functions with controlled Fourier coefficiens

It's a well known fact that an infinite dimensional Banach space $E$ is not locally compact. One may consider, at which point, is this property lost, i.e. what kind of compact sets $K \subset E$ exist....
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### Ambiguity in definition of compactness

I am struggling with the definition of compactness in a topological sense. Below is the definition presented in my lecture notes: A topological space $X$ is compact if every open cover has a ...
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### When is an Open Set Homeomorphic to the Interior of its Closure?

Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$. I am looking for known assumptions on $X$ and $U$ such that one of the following ...
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### Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
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### Stone-Cech compactification of real line

I know that $[0,1]$ and a unit circle $\mathbb{S}^1$ are one-point compactifications of $\mathbb{R}$ under some suitable homeomorphism. But how does one construct the Stone-Cech compactification?
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### $X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?

Let $X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then is it true that the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?
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### Show by using finite intersection property that( $\mathbb R$,d) is not compact.

I know that this problem is an application to the statement- ($\mathbb X$,d) is compact$\iff$Every collection of closed sets in ($\mathbb X$,d) with the finite intersection property has a non-empty ...
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### compactness of a sequence space

Sorry if this question might be not well-posed, I'm very very new to topology. I have a compact set $S$ of sequences $(x_n)_{n\in\mathbb N}$ in $\mathbb R^n$ and those sequences are bounded, in the ...
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### Compact subsets in Topology of pointwise convergence

First of all, I know a similar question has been asked here compactness in topology of pointwise convergence, but I am still do not know how to identify compact subsets. Given a set $X$, endowed with ...
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### Why Munkres §26 Exercise 11 is nontrivial?

This is probably a silly question, but I have a trivial (most likely wrong) reading of Munkres §26 Exercise 11: Let $X$ be a compact Hausdorff space. Let $\mathcal{A}$ be a collection of closed ...
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### Problem to demonstrate that it is compact

Let $X$ be a Normed Vector Space. For any $x\in X$ and $r>0$, let $W:=\{y∈X:∥y−x∥≤r\}$. Prove: $W$ is closed and if $\dim(X)<\infty$ $W$ is compact. I have no problems show that it is closed, ...
### Metric space on $\mathbb{R^n}$ where Heine-Borel criterion does not hold
Heine-Borel criterion of $\mathbb{R^n}$ : closed and bounded $\implies$ compactness Give an example of a metric space in $\mathbb{R^n}$ where this criterion does not characterize compactness ...