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 ...
3
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1answer
55 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
36 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 $K\subset V\subset\bar{V}\subset U$ ?
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1answer
51 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 ($f \in \mathcal{C}^0(\mathbb{R},\mathbb{R})$...
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1answer
29 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
306 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 sub-...
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1answer
27 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, $(1) \implies (2)$. ...
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1answer
48 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
47 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
120 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
46 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 ...
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1answer
39 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?
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1answer
50 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
46 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
741 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
57 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
198 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 $H : [0,1] \times X \to Y$...
3
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1answer
91 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 $ ...
11
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1answer
282 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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4answers
382 views

Show that a locally compact Hausdorff space is regular.

Show that a locally compact Hausdorff space $(X,\tau)$ is regular. I have already shown that a compact Hausdorff space is regular. My textbook proposes 2 methodes, but I get stuck at both. The ...
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1answer
72 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
48 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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477 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
152 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
110 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
19 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 a ...
0
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1answer
25 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 ...
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1answer
51 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
37 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 $\mathbb{R}_{\geq a}$....
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3answers
71 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?
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1answer
66 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
50 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
65 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
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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
36 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 $\delta$ ...
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0answers
15 views

Let ϵ=d(K,∂U)/2 , V=Bϵ(K), then V⊆V¯⊆U.

Let $K \subseteq U\subseteq R^{n} $, where K is compact, nonempty and U is open. Let $\epsilon = d(K, \partial U)/2$. Show that closure of set $V = B_{\epsilon}(K) $ is compact and $K \subseteq V\...
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1answer
31 views

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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1answer
90 views

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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2answers
103 views

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?
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2answers
27 views

Show that the closed ball $B[x,1]$ in $c_0$ is not compact.

Consider $c_0 = \{(a_n)_{n \in \mathbb{N}}\subset \mathbb{R}:a_n \to 0\}$. Show that the closed ball $B[x,1]=\{y \in c_0 : d_{\infty}(x,y)\leq 1\}$ is not compact in $c_0$. Were $d_{\infty}(x,y)=sup_{...
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3answers
94 views

(Non-Euclidean) Compactness

Compactness in Euclidean Space The only definition of compact set that ever made sense to me was the intro calculus one: A set is called compact if it is closed and bounded.    &...
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1answer
99 views

Question on one point compactification

I was given the following question in my general topology class assignment which is multi parts - most of which I managed alright by myself some of which I need help on. We are given a non compact ...
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1answer
128 views

prove finite intersection property for compact sets using sequential compactness

Prove finite intersection property for compact sets in metric spaces using sequential compactness with a direct proof . One approach is to prove sequential compactness and covering compactness are ...
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1answer
52 views

L^p spaces are separable and complete but not compact?

Where is the mistake in my reasoning?: Let X be a separable metric space, then for every $p\in [1,\infty)$ and for every borel measure $\mu$ on $X$: $L^p_{\mu}(X)$ is separable. Therefore by a ...
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2answers
67 views

Show that $F \subseteq X$ is closed iff $F \cap K$ is closed for every compact set $K\subseteq X$

Let $(X,d)$ be a metric space. Show that $F \subseteq X$ is closed iff $F \cap K$ is closed for every compact set $K\subseteq X$. If $F\subseteq X$ is closed then $K\subseteq X$ compact implies $K$ ...
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1answer
51 views

if one of the sets A and B is compact then d(A,B)>0.

Let $A$ and $B$ be two nonempty disjoint subsets of $\mathbb{R}^{n}$. Put $d(A,B)=inf\left \{ ||a-b||:a\in A, b\in B \right \}$. a) Show that if one of the sets $A$ and $B$ is compact then $d(A,B)&...
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2answers
98 views

Extreme value theorem, without Heine Borel.

I was wondering, if there are any mistakes, in this proof of the extreme value theorem: Theorem. Let $X$ be a compact set and $f:X\rightarrow\mathbb{R}$, s.t. $f$ is continuous. Then there exists $x\...
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1answer
233 views

Prob. 5, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: Every compact Hausdorff space is a Baire space

This is problem 5 in section 27 of Munkres' TOPOLOGY, 2nd ed Let $X$ be a compact Hausdorff space; let $\{A_n\}_{n\in \mathbb{N}}$ be a countable collection of closed sets of $X$. If each set $A_n$ ...
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2answers
183 views

Does proper map $f$ take discrete sets to discrete sets?

Suppose $f:X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Are the following results true? $1$. The map $f$ takes discrete sets to discrete sets. $2$. If $f$ is ...