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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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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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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A generalization of the Arhangelskii Theorem [migrated]

Arhangeleskii's Theorem states the following For any Hausdorff topological space $X$, $$ |X|\leq2^{\chi(X)L(X)} $$ where $\chi(X)$ is the character of $X$ and $L(X)$ is the Lindelöf degree of ...
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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
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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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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
80 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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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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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
84 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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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
33 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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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
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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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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
55 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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If X is a space in the order topology with lub. If A is closed, is A compact?

In J R Munkres section 27, there is a theorem that states that every closed interval(note not ray) in the order topology where $X$ is a set with lub property is compact. I'm wondering if $X$ is a ...
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1answer
35 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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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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Does it suffices to have a compact support together with continuously differentiability to imply compactness of a subset of functions?

Is the subset of D of $C^{1}(A,\mathbb R^{L}_{++})$ where A is a compact subset of $R^{L+1}_{++} $ and $R^{L+1}_{++} $=$({a\in\mathbb{R}^{L+1}|a>>0})$ , where D is defined as a all functions of ...
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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
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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 $E$ ...
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alternative Compactness theorem proof

I'm attempting a problem which requires me to prove the compactness theorem for propositional logic ![enter image description here][1]in a slightly different way to normal. I'm struggling to ...
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How to prove that a metric space is compact if it is complete and totally bounded?

How to prove that a metric space is compact if it is complete and totally bounded? Wiki wrote that it is a generalisation of Heine–Borel theorem but I can't prove it.
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Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
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Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
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Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
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Compact Hausdorff topological spaces

If we have a topological space $(X,\mathcal{T})$, where it is compact and Hausdorff, them we can say that any other topology $\mathcal{H}$ on $X$ such that $\mathcal{T}\subseteq\mathcal{H}$, the ...
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Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
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1answer
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Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
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1answer
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Cardinality of all compact metric spaces

I`m looking for cardinal number of all compact metric spaces. I know that: Cardinal number of compact set is at most $\mathfrak{c}$ (it is a continous image of Cantor set) Compact metric space is ...
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Compact set in $(\mathbb R,\rho_1)$

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by ...
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99 views

Property of compact, convex sets in $\mathbb{R}^3$

How to solve the following: Let $K\subset \mathbb{R}^3$ be a convex, compact set with smooth boundary $C=\partial K$ and let $\vec{u}$ be any vector. Show that there exist points $x\neq y$, ...
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Naive question about the group $SU(n)$?

As usual, let $SU(n)$ represent the set of all the $n\times n$ unitary matrices with determinant $1$. It's easy to show that any matrix $U$ takes the form $U=e^{iA}$ ($A$ is a $n\times n$ traceless ...
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1answer
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A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$.

Let $f_n \to f$ on compact subsets of the real line. If $u_m \rightharpoonup u$ in $L^2(0,T;H^1) \cap L^p(0,T;L^p)$ and $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in ...
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1answer
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Using the compactness theorem

I am working through problems which ask you to apply the compactness theorem (from propositional logic) to problems. How would you go about solving this one? Let $\mathbf{L}$ be an arbitrary ...
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Stone-Čech compactification not by ultrafilters only.

I am familiar with Stone-Čech compactification using ultrafilters. But, I, somehow can't understand the construction by commutative diagram, and certainly can not see the connection between the two ...
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1answer
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For compact subspaces $C$ and $K$ of $X$ and $Y$, prove that for every open set $U$ of $X \times Y$, there exist open sets $V$ and $W$ with…

Let $C$ be a compact subspace of $X$ and let $K$ be a compact subspace of $Y$ . Let $U$ be an open set in $X \times Y$ containing $C \times K$. Show that there exist open subspaces $V$ of $X$ ...
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Covering dimension of a compact metric space

I would like to see the proof of the following fact (references appreciated). A compact metric space $X$ has covering dimension $\leqslant n$ if and only if there is a continuous surjection $\pi ...
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compactness, real analysis

I need help with this excercise, I got a), and c must follow from the extreme value theorem, and the last answer in c is yes? My main problem is with b). It seems that I should start with an open ...
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Approximation by finite sets

I'm reading the book "Topology and Order" by L.Nachbin. In chapter $3$ he speaks about properties of compact Hausdorff spaces. He writes: [A]lthough these spaces may be infinite, they admit ...
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Is the set of translations of a function compact?

Let $X=BUC(\mathbb{R})$ be the Banach space of real bounded uniformly continuous functions on $\mathbb{R}$ equipped with the supremum norm. Let $f\in X$, then the subset $$\{f_a:t\mapsto f(t+a), \ \ ...
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How to “convert” from net to sequence in a first countable space

In a first countable space, what's a good way of going from nets to sequences? Let me explain more clearly what I mean. Suppose $f:X\to Y$ is a topological map and $X$ is first countable. Then I ...
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An infinite compact set which allows no boundedness and analyticity

I need an example of an infinite compact set $K$ in $\mathbb {C}$ such that there does not exist any non-constant function which is both bounded and analytic on $\mathbb{C} - K$. First, any hints ...
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How to show $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ is path connected and compact?

let $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ . Show that A is path connected and compact. my attempt: since $\frac {x^2}{9}+\frac{y^2}{4}=1$ is elips. A is bounded and closed. so is compact. (by heine ...
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Topological distinguishibilty of $\infty$ after one point compactification?

Let $X$ be the one point compactification of some locally compact Hausdorff space. Let $\infty \in X$ represent the added point. Is there always a homomorphism $\phi:X \to X$ with $\phi: \infty ...
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1answer
60 views

Countable union of relatively compact sets

Let $X$ be a topological space and $\mathcal K(X)$ be $\sigma$-algebra, generated by compacts of $X$. Prove that for any set $B \in \mathcal K(X)$ either $B$ or its complement can be represented as a ...
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Jordan content under continuous differentiable map

I have the following problem which seems simple but in fact I find no proof for it so I am wondering if I could get some help. Let $A$ be a compact set subset of an open set $U$ in $\mathbb{R^n}$, ...