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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1answer
29 views

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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1answer
71 views

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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1answer
73 views

Urysohn's Lemma, Stone-Weierstrass

Let $X$ be a compact space. Show that the following statements are equivalent: a) $X$ is homeomorphic to a compact subset of $\mathbb{R}^n$ b) There are functions $f_1,\dotso, f_n\in C(X)=\{...
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3answers
44 views

Homeomorphic but not equivalent compactifications.

I stumbled upon the definition of equivalent compactifications which is: Two compactifications $Z_1$ and $Z_2$ of the space $X$, are said to be equivalent if there exists a homeomorphism $h:Z_1\...
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1answer
21 views

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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2answers
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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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1answer
47 views

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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1answer
38 views

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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25 views

compactness of thes sequence set

Let $S$ be a compact (in the usual topology) subset of $\mathbb R^n$, let $W = \{(q_k)_{k\in\mathbb{N}}\,\mid\, q_k\in S\}$ be the set of all the sequences taking elements in $S$, let $(f_k)_{k\in\...
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0answers
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"Correct'' morphism extension to Nagata compactifications

Can a morphism of separated schemes of finite type over a field be extended to Nagata compactifications of the schemes preserving the closed complements? Let $\mathbf{Sch}/k$ be the category of ...
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1answer
29 views

Exercise I.7.2 in Geometry and Topology by Bredon

I'm working though the first chapter in Geometry and Topology by Glenn Bredon, and I'm stuck on Exercise I.7.2, which is related to compactness. It reads: Let $X$ be a compact space and let $\{C_{\...
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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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1answer
71 views

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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1answer
37 views

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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0answers
13 views

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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4answers
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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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3answers
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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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2answers
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Compactness of infinite union under these conditions

Assume I have an infinite sequence $(S_k)_{k\in\mathbb N}$ of sets $S_k\subset \mathbb R^n$, assume that all the $S_k$ are compact with respect to the topology induced by some metric $d:\mathbb R^n\...
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1answer
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Show that for all compact $K$ and for all continuous function $f:X \to K$, there is $g: \overline{e(X)} \to K$ continuous with $g \circ e = f $.

Let $X$ a Tychonoff space, $S = ${$f:X \to [0,1] : f$ continuous} and consider the topology immersion $e: X \to \prod_{s \in S} [0,1]$ where $e(x) = (f(x))_{f \in S}, \quad \forall x \in X$. Show ...
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2answers
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Does there exist a compact metric space $X$ containing countably infinitely many clopen subsets?

From this Clopen subsets of a compact metric space we know that any compact metric space $X$ contains at most countably many clopen subsets ; my question is : Does there exist a compact metric space $...
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2answers
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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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1answer
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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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0answers
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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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1answer
31 views

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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2answers
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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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1answer
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Determine whether the differential operator is compact in the following cases

Given the differential operator $\displaystyle Tx(t)=\frac{dx}{dt}$, I need to determine (and be able to justify) whether it is compact in the following three cases: $T: C^{1}[0,1]\mapsto C[0,1]$ $T:...
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1answer
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Issue with compactness implies boundedness proof

The proof is outlined as follows: (copied from Wikipedia but Apostol gives the same idea) If a set is compact, then it is bounded: Consider the open balls centered upon a common point, with any ...
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1answer
45 views

Prove the integers in the arithmetic progression topology is not compact

I've been studying for my final exam in a general topology course, and I came upon this problem about compactness that I'm have a really tough time solving. Let $a$ and $b$ be integers, with $b\neq 0$...
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1answer
36 views

Compactness in a vector space

If $E$ is a normed space and $F$ is a subspace of $E$, how to prove that if $F\neq\{0\}$ then $F$ is not compact? I begin by this let $x\in F$ then $F=\bigcup_{x\in F} B(x,\varepsilon)$ how to say ...
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1answer
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If $X$ is a non-compact metric space, can $X^n$ ever be compact?

Do there exist metric spaces $X$ such that $X^n$ is compact even though $X$ is not? Since compact spaces can have non-compact subspaces, e.g. $[0,1)\subset[0,1].$
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1answer
49 views

Compactification, definite function

Let $\hat{X}$ be the compactification of a Locally compact Hausdorff-space $X$. Show, that it exists an unique, continuous function $p_{\hat{X}}:\hat{X}\to X^+$, whose restriction on $X$ is the ...
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2answers
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compactness of a set of sequences

I'm sorry if this is probably a stupid and not well-posed question but I'm really new to topology. I have two compact sets $U\subset \mathbb R ^n$ and $Y\subset \mathbb R$. Then I define $Q$ to be the ...
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3answers
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Hausdorff of $X$ implies Hausdorff of $Y$ under some strange condition

Let $p:X\to Y$ be continuous surjective closed mapping s.t. $p^{-1}(y)$ is compact $\forall y\in Y$, prove that: (a) If $X$ is Hausdorff, then $Y$ is Hausdorff (b) If $Y$ is compact, then $X$ is ...
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0answers
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Composition Series Analogous to Compactness?

The wikipedia page for group with operators makes the following claim about composition series as being analogous to compactness: The Jordan–Hölder theorem also holds in the context of operator ...
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1answer
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Proof or definition of compactness in lecture notes?

I am baffled with what I am seeing. First, here's what is noted as a definition in my notes Let $X$ be a set and $A \subseteq X$. A cover of $A$ by subsets of $X$ is a family $(W_i)_{i \in I}$ of ...
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2answers
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Compactness theorem, propositional calculus

Please help me with this problem. Prove that if $\land \Phi \models \lor \Psi$ (both $\Phi$ and $\Psi$ infinite) then there exist $\phi_1,...,\phi_n$ from $\Phi$ and $\psi_1,...,\psi_m$ from $\Psi$ ...
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Characterization of Compact Space via Continuous Function

Let $(X,\mathfrak{T})$ be a topological space. We know that if $X$ is compact and $f:X\to \mathbb{R}$ be any continuous function then $f(X)$ is bounded since the continuous image of a compact set is ...
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2answers
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Prove all closed subspace of a compact space are compact: Redundancy?

I see a redundancy in the following proof of the statement. First, we have a lemma that this proof uses A subspace $A \subseteq X$ is compact if and only if every open cover of $A$ by open subsets ...
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1answer
49 views

Three notions of compactness: minimal conditions for equivalence?

There exist 3 notions of compactness: $X$ is compact if any open cover admits a finite subcover; $X$ is sequentially compact if any sequence in $X$ has a convergent subsequence; $X$ is limit point ...
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1answer
18 views

Partial converses to extreme value theorem

Under what conditions can we establish a converse to the extreme value theorem? That is, for what topological spaces $(X, \tau)$ can we say that if $(\forall f \in C(X))(\exists c \in E) \left( f(c) = ...
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1answer
15 views

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, ...
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2answers
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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 ...
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58 views

How to prove a set is closed

Let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a finite probability space equipped with a filtration, i.e an increasing sequence of $\sigma$-algebras included in $\mathcal{F}$ : $\mathcal{F}_0, ...
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1answer
35 views

Compactness and sequences in $\mathbb{R}^{n}$

Why is it that: If $A$ is a compact set and $\left ( a_{n} \right )$ a sequence in $A$, then there is a subsequence $\{a_{n_k}\}$ such that $\lim_{k\to\infty} a_{n_k}=a$ with $a\in A$. I get ...
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1answer
25 views

Prove the boundary is a compact 1 manifold

A closed surface with boundary is a compact connected topological space $B$ with the property that each point $p \in B$ has an open neighborhood $U$ homeomorphic to either: $\{(x, y) \in \mathbb{R^2}|...