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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27 views

Open (closed) sets of a locally compact space.

Let $X$ a locally compact space. How do I show that if $A$ is a open (closed) set in $X$ then $A$ is locally compact? Thank you very much.
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1answer
24 views

Distance of a point to a subset.

Let $(M,d)$ be a metric space. For a subset $A\subseteq M$ we define the distance of a point $x$ to $A$ as $$\alpha_A(x):=\operatorname{dist}(x,A):=\inf_{y\in A}d(x,y)$$ Prove that: ...
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3answers
72 views

Which of $(-\infty,\infty]$ and $[-\infty,\infty]$ is homeomorphic to $S^1$?

Is it correct to say that $(-\infty,\infty]$ is homeomorphic to $S^1$? or it is $[-\infty,\infty]$? (considering standard topology). Would you please provide some explanation or better a rigorous ...
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0answers
55 views

Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
-1
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1answer
53 views

Find the limit of $A={\{(\dfrac{\theta-1}{\theta}}, \theta)|\theta=1,2,3,\dots\}$

Question: Find the limit of $A={\{(\dfrac{\theta-1}{\theta}}, \theta)|\theta=1,2,3,\dots\}$? Here, $\left(\dfrac{\theta-1}{\theta},\theta\right)$ is a point in $\mathbb R^2$ expressed in polar ...
2
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1answer
23 views

Integral of Laplace-Beltrami operator over a manifold

Consider an equation $$\Delta u=-he^{u}$$ over a compact 2-manifold $M$, where $u\in C^{\infty}(M)$. In paper "Curvature functions for Compact 2-Manifolds" by Kazdan&Warner it is said that ...
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1answer
63 views

Brouwer fixed-point theorem infinite dimension [closed]

Brouwer fixed-point-theorem holds for compact convex set. Do you have example(s) where the theorem doesn't hold in infinite dimensional Banach spaces?
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2answers
390 views

Why we use the word 'compact' for compact spaces?

Considering the definition of compactness in either Analysis or Topology books, or its equivalent definitions (i.e. [It] is compact $\Longleftrightarrow\dots$), I couldn't understand why ...
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0answers
21 views

countable dense set of space of continuous functions on a campact set

Let $X$ be a compact metric space. Let $C_+(X)$ be the set of all non negative continuous functions on $X$. Do there exist a countable dense set of $C_+(X)$? I think the answer is affirmative. For ...
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0answers
39 views

closedness of compact sets in some topological spaces

Is there any famous axiom on X other than Hausdorffness or axioms leading to Hausdorffness,such that every compact set in X is closed?
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1answer
14 views

Infimum of the supremum absolute value of a decreasing sequence of subsets of $\mathbb{C}$ with non-empty intersection

Let $K_{n}$ be a decreasing sequence of bounded subsets of $\mathbb{C}$ such that $\cap_{n}K_{n}=K\neq\emptyset$. Let $\lambda_{n}=\text{sup }_{\lambda\in K_{n}}|\lambda|$ and $\lambda_{0}=\text{sup ...
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1answer
73 views

Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: Why we need continuity to show the result?

Let $f: X\mapsto Y$ be a closed continuous surjective map such that $f^{-1}(y)$ is compact, for each $y\in Y$. Show that if $Y$ is compact, then $X$ is compact. My question is why do we need $f$ to ...
3
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2answers
32 views

What is “approximate compactness”? What is an example of an approximately compact set?

I read this: A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a ...
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1answer
49 views

An simple example to show that every countably compact space needn't be compact

I am willing to study compact and connected in topological space and apply in other topological spaces. I am a beginner in this subject. Kindly give some examples. I have went through few books but I ...
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1answer
19 views

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 ...
3
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1answer
50 views

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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1answer
25 views

Why is this image sequentially compact? [duplicate]

Assuming $X$ and $Y$ are normed spaces, $K\subset X$ and $f:K\rightarrow Y$. Why is the image $f(K):=\{f(x)\in Y: x\in K\}$ sequentially compact, if $K$ is sequentially compact and $f:K\rightarrow Y$ ...
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1answer
32 views

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 $K\subset V\subset\bar{V}\subset U$ ?
2
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1answer
41 views

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 ...
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1answer
23 views

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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1answer
37 views

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 ...
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1answer
22 views

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 ...
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1answer
40 views

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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1answer
26 views

For compact $A$, $\inf\{\varrho(y,x) : y \in A\}=\varrho(a,x)$

I need help with prooving that if non empty $A$ $\subset(X,\varrho)$ is compact, then: $(\forall x \in X) (\exists a \in A) \inf\{\varrho(y,x) : y \in A\}=\varrho(a,x) $ I found this solution: ...
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1answer
99 views

Alternate proof for Arzela-Ascoli

Im trying to finish a beautiful excercise, which consist of giving an alternate proof for the following corollary of Arzela-Ascoli´s Theorem. Given $X,Y$ metric spaces, $X$ compact, $Y$ complete, and ...
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1answer
22 views

Have you ever seen this result about pointwise/uniform convergence of a net of continuous functions?

I am in need of results transforming pointwise convergence of functions into uniform convergence. Since I wasn't satisfied with Dini's theorems, I had to prove the following result: Let $K$ be a ...
0
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1answer
23 views

Can a countably infinite compact topological space have isolated point? Can it admit a minimal subsystem?

Examples I could think of are all sequences with their limit. But is every countably infinite compact space admit atleast one isolated point?
2
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1answer
22 views

intersection of two relatively compact spaces

It is known that intersection of two compact spaces is might not compact but intersection of two compact Hausdorff spaces is compact. I curious about intersection of two relatively compact spaces. In ...
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0answers
28 views

Countable fundamental system of neighbourhoods in a compact Hausdorff space?

Is it true (or false) that every point in a compact Hausdorff-Space has a countable local base, i.e. a countable fundamental system of neighbourhoods? If this is false, which additional property ...
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2answers
371 views

Topological spaces in which every proper closed subset is compact

Let $X$ be a topological space. It is a basic result that that if $X$ is compact, then every proper closed subset $Y \subset X$ is compact. Out of curiosity, I would like to explore the converse of ...
2
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1answer
46 views

Comparing the Samuel and Stone-Čech compactifications of a Hausdorff topological group

Let $G$ be an Hausdorff topological group and let $\beta G$ be the Stone-Čech compactification of $G$. Now, $G$ is also a uniform space with respect to the so-called right uniformity. Let $S(G)$ be ...
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2answers
125 views

Is $\mathbb{R}^n$ properly homotopy equivalent to $\mathbb{R}^m$ if $n \neq m$?

$\DeclareMathOperator{\id}{id} \newcommand{\R}{\mathbb{R}}$ If $f,g : X \to Y$ are two maps (all maps considered are continuous here), a homotopy between $f$ and $g$ is a map $H : [0,1] \times X \to ...
3
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1answer
62 views

Finding Function's Extension and Its Unique Existence.

Let $$A= \left\{\frac j{2^n}\in [0,1] \mid n = 1,2,3,\ldots,\;j=0,1,2,\ldots,2^n\right\} $$ and let $$ f:A\rightarrow R $$ satisfy the following condition: There is a sequence $ \epsilon_n \gt 0 $ ...
12
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1answer
228 views

Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected.Then show that $\partial D$ is connected

I am trying to prove the following famous result in Point Set Topology. Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected. Then show that ...
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1answer
46 views

Extreme points of the set of positive regular borel measures on a compact Hausdorff space

I have some troubles with a specific proof of a (Bochner-type) theorem in Rudin's book "Functional Analysis". More specifically, let $X$ denote a compact Hausdorff-Space and let $M$ denote the set of ...
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2answers
40 views

Why $\hat{\mathbb{C}}\setminus K$ connected $\implies {\mathbb{C}}\setminus K$ connected? ($K $ compact)

Let $\hat{\mathbb{C}}=\mathbb{C}\cup \{\infty\}$ denote the extended complex plane, with the usual topology.That is $U$ such that $U$ is open in $\mathbb{C}$ and the neighbourhoods of $\{\infty\}$ ...
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3answers
319 views

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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0answers
90 views

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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0answers
9 views

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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1answer
62 views

Which of the following condition implies that the set $A$ is compact

Question : Let $A$ be a subset of $\mathbb R$. Which of the following properties implies that $A$ is compact $?$ Every continous function $f :A \rightarrow \mathbb R $ is bounded. Every sequence ...
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1answer
14 views

Prove that there exist $r>0$ such that $\bigcup_{x \in K} B(x,r) \subset V$

Let $M$ be a metric space, let $K \subset V \subset M$, $K$ compact, $V$ open. Prove that there exist $r>0$ such that $\bigcup_{x \in K} B(x,r) \subset V$ I came up with a proof, but there is ...
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1answer
14 views

Distance attained by a function

Let $A$ be a subset of $\mathbb R^n$ and let $x\in \mathbb R^n$. Then $\exists y_0\in A$ such that $d(x,y_0)=d(x,A)$ if $A$ is a non-empty subset of $\mathbb R^n$. $A$ is a non-empty closed subset ...
0
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1answer
49 views

Is Y a compact subset?

$X$ compact topological space, $f\colon X\to X$ continuous Is then $Y:=\bigcap_{n\geqslant 1}f^n(X)\subset X$ compact? Edit (based on the comments I got below): The assumption that $X$ ...
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1answer
34 views

How can I formally write $f(x) \to 0$ when $x \to \infty$

I've just proven that if $f:\mathbb{R} \to \mathbb{R}$ is uniformly continuous in $[a,b]$ and it is also uniformly continuous in $[b,+\infty)$ then $f$ is uniformly continuous in $\mathbb{R}_{\geq ...
2
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3answers
58 views

A compact Hausdorff space

It is known that every finite space is compact. Then I am worried whether there exists a compact Hausdorff space $X$ with with ordinal of $X$ is $\omega_0$. Does anyone know about it?
3
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1answer
51 views

$f\circ g$ continuous, $f$ local homeomorphism, $g$ continuous in a different topology $\implies g$ is continuous

I've asked this question before but neglected some assumptions and got a less than useful answer as a result, so I'm going to try again. Let $g:I\times I\to Y$ (where $I=[0,1]$) be a function such ...
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1answer
42 views

Condition under which a set is compact

I'm studying at university real analysis and in class the teacher said that a set is compact if and only if is closed and bounded. But I don't really understand the concept, more widely: what really ...
0
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1answer
61 views

Can someone please point out the flaw in my proof? [duplicate]

Let $f:X \to Y$ be a proper map.Show that $f$ takes discrete sets to discrete sets. Proof:Let $A$ be discrete in $X$ and let $K$ be compact in $Y$ then $f(A) \cap K=f(A \cap f^{-1}(K))$,is finite ...
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0answers
19 views

Upper-hemicontinuity of product maps on compact metric spaces.

Let $X$ and $\{Y_i\}_{i\in I}$ be compact metric spaces (where $I$ an index set of possibly uncountable cardinality). Let $\Gamma_i$ be a compact valued, upper hemicontinuous (UHC) correspondence from ...
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2answers
32 views

Prove that $\delta$ is a metric in $\mathcal{K}(X)$

Let $(X,d)$ be a complete metric space. We define $\mathcal{K}(X)=\{K \subset X : K \text{ is compact and non empty}\}$ Define $d'(A,B)=sup_{a \in A}\{d(a,B)\}$ Show that $\delta$ ...