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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Compact sets and Kuratowski limit

I have been struggling with the following claim: Let $A_n$ be a sequence of compact sets and $A$ a compact set. $A=\lim\sup_n A_n=\lim\inf_n A_n$ iff $d_H(A_n,A)\to 0$ where $d_H(.,.)$ is the ...
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0answers
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Isomorphism isometries between finite subsets , implies isomorphism isometry between compact metric spaces

Let's $(X_1,d_1), (X_2,d_2)$ be compact metric spaces such that for every finite subset of $X_1$ like $A$ (respectively any finite subset of $X_2$ like $B$ ) there exists a finite subset of $X_2$ ...
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1answer
39 views

Alexandroff one-point compactification = Freudenthal end point compactification / set of ends

By spaces I mean locally compact, $\sigma$-compact, connected, locally connected, Hausdorff topological spaces. I need a reference (not a proof, I already have one -- or at least I think so; ...
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3answers
104 views

Compactness under different metric?

Consider the metric $\rho(x,y)=\frac{|x-y|}{1+|x-y|}$ on $\mathbb{R}$. Is $(\mathbb{R},\rho)$ compact? In order to show that is not, I wanted to find a sequence such that any subsequence is ...
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Confusion in proof of theorem ($2.7$) in Rudin's Real and complex analysis

I am not able to fill the gap in proof of following theorem which is stated as... Let $U$ be an open set in a locally compact hausdorff space $X$, $K\subset U$ and K is compact. Then there exists an ...
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1answer
36 views

Compactness and Arithmetic Confusion

Let $T$ be some theory capable of arithmetic and construct a provability predicate (which we will call $Prb_T$). Let $\mathbb{N} \models T$. Expand our language to include a new constant symbol $c$. ...
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83 views

Suppose $K$ is compact. What other types of coverings must have finite subcovers?

Let $X$ be a topological space. Call $S\subseteq X$ an $\mathcal{O}$-set if there exists an open set $O$ such that $O\subseteq S \subseteq \overline{O}$. Suppose $X$ is compact. Is it true that any ...
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1answer
198 views

Stone Cech compactification homeomorphism implies realcompactification homeomorphism

I was wondering: If $\beta X$ is homeomorphic to $\beta Y$, is it true that $\nu X$ is homeomorphic to $\nu Y$? Notation: If $f: X\rightarrow \mathbb R$, we denote it's extension by $f^\alpha: \beta ...
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2answers
93 views

Continuous function from R to a compact set

I know that a continuous function maps compact sets into compact sets. My question now is, are there continuous functions $f:{\mathbb R}\rightarrow I$, with $I=[a,b]$ ($a\neq b$)?
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1answer
34 views

zero-sets of $\beta X$

I'm trying to understand the following proof from Walker: Proposition. The zero-sets of $\beta X$ are countable intersections of closures in $\beta X$ of zero-sets of $X$. Proof. If $Z$ is a zero ...
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16 views

Converse to sequential Banach--Alaoglu [duplicate]

Let $B$ be the closed unit ball of the dual space of a real normed vector space $V$. If $V$ separable then $B$ is sequentially compact in the weak-* topology. What about the converse?
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1answer
45 views

Prove that a set is compact

Let $X$ be a compact space, let $U$ be an open set in $X$, Let $f:U\to [0,1]$ be a continuous map. Prove that the set $$K=\{(x,t): x \in U , 0 \leq t \leq f(x) \} \subset X \times [0,1]$$ is compact. ...
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27 views

Uniform convergence on compact sets allows switching the limit and the integral.

Why does uniform convergence on compact sets allows switching the limit and the integral?
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5answers
83 views

If $A$ and $B$ are compact subset of $\mathbb R$ , then so is $A+B$.

Prove the following: If $A$ and $B$ are compact subset on $\mathbb R$ , then so is $A+B:= \{a+b\mid a\in A ,b\in B\}$. I was actually thinking about first proving that if $A\subseteq \mathbb R$ is ...
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2answers
71 views

Construction of an embedding of $\mathbb{Z} \cup \{\infty\}$ into $\mathbb{R}$.

Let $X$ be the one-point compactification of the integers $\mathbb{Z}$, construct an embedding of $X$ into the reals $\mathbb{R}$. I already appreciate your hints/answers. Thanks
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19 views

Are (certain) metric-preserving vector bundle maps proper?

Given two real vector bundles $p\colon U \to X$ and $q\colon V \to Y$ with a metric and a vector bundle map $f\colon U \to V$ preserving this metric (i.e. it's fiberwise an orthogonal map). Can we ...
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214 views

How do I turn my verbal argument into something formal in [Real Analysis]? (proving every compact set is bounded)

So one of the exercises I am doing is to prove (or disprove) that 'Every compact set on a metric space is bounded'. Verbally, I can 'prove' this by simply stating: "If the every compact set on a ...
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1answer
19 views

Closed kernel in a compact group is open

The way I think it should work is that $${\rm ker} = \bigcap_{g \notin {\rm ker}} (G - g\,{\rm ker}),$$ with each $G - g\,{\rm ker}$ open. Since $G$ is compact, there should, in fact, only be ...
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24 views

Definition of a Paracompact space

I have a question about the definition of a paracompact space. We said that a space $X$ is paracompact iff $X$ is $T_2$ and if any open covering of $X$ has a finer locally-finite covering. I don't get ...
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1answer
50 views

Choice of number in the proof the 5-r covering theorem

Why has the number 3 been chosen? I have tried drawing this and it seems wrong (its not). The balls definitely dont seem to be disjoint either. It would seem that if a particular $x$ has $r(x)$ ...
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1answer
50 views

The definition of Compactness for “set” and “space”

Compactness for "set" and "space" I was wondering if there is any significance between the two settings. Do we treat them as two different things? For example, let $(X,d)$ be a metric space with the ...
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0answers
92 views

If $X$ is compact and $f:X \rightarrow Y$ is a dense continuous injection, then $f$ is a homeomorphism

I found this: Let $X$ be a compact space and $f:X \rightarrow Y$ a continuous injection. Let $f(X)$ be dense in $Y$. Prove that $f$ is a homeomorphism. So, my question is: is it possible to prove ...
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A metric on $\mathbb{N}$

Define a metric on $\mathbb{N}$ by fixing a prime, $p$, and setting $$d(x,y)=\begin{cases} 0 & x=y \\ p^{-k} & \text{otherwise} \end{cases}$$ where $p^k$ is the highest power of $p$ that ...
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1answer
74 views

Are the family of given nice functions $f\subset C^0(I,[0,1])$ equicontinuous?

The family of continuous, bounded (in $[0,1]$) functions $f\in\mathcal{F}$ are defined on a closed subset of real numbers $I\subset\mathbb{R}$ as follows: \begin{equation} f(y) = \begin{cases} 0, ...
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1answer
52 views

Are the family of functions $C^0(I,[0,1])$ equicontinuous?

I searched but couldn't find. Are the family of continuous functions $C^0(I,[0,1])$ equicontinuous for the finite interval $I\subset\mathbb{R}$? To claim this, I guess for every $\epsilon>0$ ...
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3answers
65 views

Proof that the continuous image of a compact set is compact [duplicate]

Let $X\subset \mathbb R^{n}$ be a compact set, and $f :\mathbb R^{n}\to \mathbb R $ a continuous function. Then, $F(K)$ is a compact set. See, I know that this question may be a duplicate, but the ...
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1answer
50 views

Closure and compactness of the set of real eigenvalues ​​of a real matrix.

Let $A$ be a part of $\mathcal{M}_n(\Bbb{R})$ and $B$ the set of real eigenvalues ​​of the matrix $A$. 1) Show that if $A$ is compact then $B$ is compact as well. 2) If $A$ is closed ...
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1answer
58 views

A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
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1answer
41 views

Are Hausdorff compactifications of a Tychonoff space $X$ in one-to-one correspondence with completely regular subalgebras of $BC(X)$?

Let $X$ be a completely regular (Tychonoff) topological space. It is known that if $\mathscr F\subseteq C(X,[0,1])$ separates points and closed sets (that is, for every closed set $E\subseteq X$ and ...
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57 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
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Sequence of compact sets

Let $(X,d)$ be a metric space and consider an increasing sequence $A_n$ of its subsets such that $A = \bigcup_n A_n$ is compact. Can it happen that $A\setminus A_n$ is compact for all finite $n$?
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3answers
79 views

Show that a map with some properties is closed

Let X be a topological space and Y hausdorff and local compact. Let $f:X \rightarrow Y$ be a continuous map such that $f^{-1}(K)$ is compact for all compact sets $K$. Show that $f$ is a closed map. ...
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2answers
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If X is local compact, then it holds: A is closed $\iff$ $A\cap K$ is compact for all compact K [closed]

Prove: Show that for every local compact space X holds the following: A $\subseteq$ X is closed $\iff$ $A \cap K$ is compact, for all compact sets K. I use the following definition of local ...
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1answer
39 views

The set of zeros of a holomorphic function is finite in compact sets

Statement Let $f:\mathbb \Omega \to \mathbb C$ be a holomorphic function, $f \neq 0$ ($\Omega$ is a region, i.e., an open, nonempty, connected set). Prove that in every compact subset $K$ of ...
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1answer
49 views

Closed subsets of $\beta \mathbb R$

Definitions. Suppose $X$ is a topological space. $w(X)=\min\{|\mathcal B|:\mathcal B$ is a base for $X\}+\omega$ $e(X)=\sup\{|D|:D\subseteq X$ is closed and discrete$\}+\omega$ $K(X)$ is the ...
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1answer
99 views

Topological counterexample: compact, Hausdorff, separable space which is not first-countable

I need an example for a compact, Hausdorff, separable space which is not first-countable. I tried to look for it for some time without success...
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59 views

In a metric space a compact set is closed

I want to show the following: Let $X$ be a metric space. Show that every compact subset $Y$ of $X$ is closed. The idea is to show that $X\setminus Y$ is open. So, for any $x \in X\setminus Y$, I ...
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3answers
56 views

Intuition Behind Compactification

I'm heading into my second semester of analysis, and I still don't have a good intuition of when a set is compact. I know two definitions, covering compact and sequentially compact, but both of those ...
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1answer
131 views

How to check that finite sets are dense in exp(X)?

How i can check if finite set $\bigcup F_{n}$ is dense in $exp(X)$, where $exp(X)$ is $$exp(X)= \{ A\in X ; A\not= \emptyset ; A \textit{ compact in } X\} $$ ($exp(X)$ is hyperspace, so it is set ...
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1answer
36 views

Topological property: set-theoretically large subsets of an infinite space are not compact.

Let $X$ be an infinite topological space. Say that $X$ satisfies # if no subset of $X$ of cardinality $|X|$ is compact. So for instance it is clear that no (infinite) compact space satisfies # any ...
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1answer
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Stone-Cech compactification: clarification of definition

When defining Stone-Cech compactification we take a Tychonoff space $X$, the space $C_b(X)$ of bounded continuous real functions on $X$, define $I_f$ as closed limited intervals containing $f(X)$ for ...
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1answer
44 views

Does paracompact Hausdorff imply perfectly normal?

That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially ...
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1answer
29 views

let $A$ be a subset of $\mathbb R$ s.t. both $A$ and $\mathbb R-A$ is dense in $\mathbb R$. > Then show that $A$ is nowhere locally compact.

let $A$ be a subset of $\mathbb R$ s.t. both $A$ and $\mathbb R-A$ is dense in $\mathbb R$. Then show that $A$ is nowhere locally compact.
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1answer
72 views

A set is compact if and only if every continouos function is bounded on the set?? [duplicate]

I was asked to prove the following statement: let $K \subseteq R^n$. show that $K$ is compact (meaning closed and bounded) if and only if every continouos function is bounded on $K$. What I did: ...
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2answers
30 views

Proving compactness of the extended complex plane

Prove that $(\overline{\mathbb C}, \overline{d})$ with $\overline{d}(z,z')=d(\phi(z),\phi(z'))$, where $d$ denotes the euclidean distance in $\mathbb R^3$ and $\phi$ is the inverse of the ...
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1answer
57 views

Compactness of a set of partitions

The interval $[0,1]$ is partitioned to $n$ disjoint parts. Is the set of all possible partitions compact? There are several cases: A. All $n$ parts are connected intervals (possibly empty). In this ...
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1answer
38 views

compactness and maximal elements

Let $C$ be a nonempty compact subset of $R^n$, with a certain partial order defined on it. I am trying to prove that $C$ contains a maximal element. My idea is: start with a certain element of $C$. ...
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1answer
54 views

Compactness and existence of Pareto-efficient cake partitions

I am trying to understand a fundamental statement in the theory of cake-cutting. BACKGROUND: There is a certain "cake" $C$ (a subset of $R^n$). The cake is divided among two agents, 0 and 1. Each ...
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1answer
48 views

In a compact space, every net has a convergent subnet

I'm just learning how to work with nets. I'm attempting the proof that $X$ compact $\implies$ every net in $X$ has a convergent subnet, and I wonder if I'm overcomplicating it. Suppose $\langle x_i ...
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1answer
17 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...