Questions tagged [compactness]

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. This includes logical compactness.

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If $X \subset \mathbb{R}^n$ is compact then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact - proof verification

Claim: If $X \subset \mathbb{R}^n$ is compact, then $X \times X \subset \mathbb{R}^n \times \mathbb{R}^n$ is compact. Key definitions: A set $X$ is closed if every convergence sequence converges to ...
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How to show that the Alexander Subbase Theorem is ZF-equivalent to the Compactness Theorem for first order logic?

Alexander Subbase Theorem (ASB): Let $X$ be a topological space. $X$ is compact if and only if there is a subbase $\mathcal{B}$ for the topology of $X$ such that every subcollection of $\mathcal{B}$ ...
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Convex compact set in $\mathbb{R}^n$ where, given any point in it, the result of replacing two of its coordinates with their mean lies in the set.

Let $X$ be a nonempty compact convex subset of $\mathbf{R}^n$. Suppose this subset has the following property: for every $x = (x_1, \dots, x_n) \in X$, for every $1 \le i< j \le n$, $$({x_1}, \...
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Locally compact and perfect map

Suppose $f:X \rightarrow Y$ is a perfect map between topological spaces $X$ and $Y$, i.e. $f$ is continuous, surjective and closed. Also suppose that the fibers of $f$ are compact. If $Y$ is locally ...
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Prob. 14, Sec. 30, in Munkres' TOPOLOGY, 2nd ed: The product of a Lindelof space and a compact space is a Lindelof space

Here is Prob. $14$, Sec. $30$, in the book Topology by James R. Munkres, 2nd edition: Show that if $X$ is Lindelof and $Y$ is compact, then $X \times Y$ is Lindelof. Here is my Math Stack Exchange ...
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Compactness of $\mathbb{R^2}$ and its relation between open balls

Show that $\mathbb{R^2}$ is not compact. My Attempt: First of all, I have tried two ways. First one is very short: i-) $\mathbb{R^2} \cong ((0,1),\;(0,1)) \cong ((0,\infty),\;(0,\infty))$. Therefore, ...
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Is $ \{A\in \mathbb{R}^{n,n}: 0<\det(A)<2\} $ compact?

I have the set $ K:=\{A\in \mathbb{R}^{n,n}: 0<\det(A)<2\} $ in $ (\mathbb{R}^{n,n},\|.\|_F) $. I don't know how to show if this set is compact or not compact in $ (\mathbb{R}^{n,n},\|.\|_F) $. ...
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Non compact group of quadratic matrices [closed]

Currently, I am working through Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. Right in the first Lie Group Problem (p.73) it is asked to prove that the orthogonal ...
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With X compact space and f:X→X is continuous function, prove that f(X) is compact space [duplicate]

It is look obvious but I'm having turbels proving that With X compact space and f:X→X is continuous function, than f(X) is compact space. Any ideas? Thank you!
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Compact supports of initial data implies compact supports for all $t$ in semi-linear wave equation $-\frac{\partial^2}{\partial t^2}u+\Delta u=u^3$.

Problem: Let $u(t,x,y)$ be a smooth real function defined on $\mathbb R \times \mathbb R^2$ where $t \in\mathbb R$ and $(x,y) \in\mathbb R^2$. We assume that it solves the following semi-linear wave ...
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Some questions in the compactness of Sobolev space and Holder space

Let $B_1$ be the unit ball in $\mathbb{R}^2$. Define $A=\{u \in W^{1,2}(B_1) : \|u\|_{ W^{1,2} } \leq 1\}$, $B= \{u \in L^{2}(B_1) : \|u\|_{ L^2 } \leq 1\}$, $C=\{u \in C^{0,\alpha}(B_1) : \|u\|_{ ...
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Approximating minimizers with $\epsilon$-nets

Let $K$ be a metric space. Fix $\epsilon > 0$. We say that $E \subset K$ is an $\epsilon$-net of $K$ if for each $x \in K$ there exists $e \in E$ such that $d(x,e) < \epsilon $. Consider a ...
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Not a locally compact space that can be represented as union of two locally compact spaces (open and close) [R. Engelking, exercise 3.3.C]

Define a subspace of the real line that can be represented as the union of two locally compact subspaces, one of which is closed and the other open, and that is not a locally compact space.
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Ideals of $C^0([0, 1]; \Bbb R)$ and compactness [duplicate]

Let $C := C^0([0, 1]; \Bbb R)$ the ring of continuous real functions on $[0, 1]$. Let $I \subset C$ an ideal. We suppose that $I$ is not contained in any $I_x:= \{f \in C \lvert f(x) = 0\}$. Show that ...
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Exercise 12, Section 26 of Munkres’ Topology

Let $p:X\to Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact, for each $y\in Y$. (Such a map is called a perfect map) Show if $Y$ is compact, then $X$ is compact. [Hint: ...
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Connected complement of compact subset

Let $U$ be an open subset of $\mathbb{R}^n$ and $K$ a compact subset of $U$ such that $U\setminus K$ is connected. Does there exist an open set $V$ such that $K\subseteq V \subseteq \overline{V} \...
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2 answers
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what is compactness argument

I'm working on this paper and I don't know what is meant by compactness argument in the proof of corrollary 4 page 226 which said that: the function $\lambda \to \|B-\lambda A\|$ (where A and B are ...
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4 votes
2 answers
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A version of Brower's fixed point theorem for contractible sets?

Brouwer's fixed point theorem states that a continuous map $f:B^n\to B^n$ ($B^n\subset\Bbb R^n$ being the $n$-dimensional ball) has a fixed point. It is clear that we can replace $B^n$ with a space $X$...
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Showing that K-topology is not path-connected without using compactness

Let $\mathbb{R}_K$ denote the real line in the K-topology, which is the topology generated by the basis $\left\{(a,b)|a,b \in \mathbb{R}\right\} \cup \left\{(a,b)-K|a,b \in \mathbb{R}\right\}$, where $...
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How to generalize Riesz–Markov–Kakutani representation theorem from $C_c(X)$ to $C_0(X)$?

I'm reading about Riesz–Markov–Kakutani representation theorem from this page. Let $X$ be a locally compact Hausdorff space. $C_c(X)$ the space of continuous compactly supported complex-valued ...
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Locally compact Polish space is $\sigma$-compact?

I have recently encountered this result. Let $X$ be $\sigma$-compact, locally compact Hausdorff space and $\mu$ is a Radon measure on $X$. Then the space of continuous functions with compact support ...
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The Stone-Cech compactification of a countable disjoint union of closed intervals

Let $X$ be the disjoint union of a countably infinite number of copies of $[0,1]$. Is $\beta X$ strictly bigger than $\beta{\bf N}\times [0,1]$? I am thinking of a bounded sequence $(f_n)$ of ...
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4 votes
1 answer
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Where do I use finiteness in this proof of: In ZF, the compactness theorem implies the Axiom of Choice for collections of finite sets?

Work in ZF, and assume the compactness theorem. Let $\mathsf{AC}^\text{fin}$ be the sentence "every collection of finite non-empty sets has a choice function". UPDATE: Thank you to the ...
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A (sub)neighborhood $S'$ of a set $X^*$ such that the line segment connecting any point in $S'$ and its projection to $X^*$ is contained in $S'$.

Consider a closed set $X^* \subset \mathbb{R}^n$. Let $Proj_{X^*}(x)$ denote the set of metric projections of $x \in \mathbb{R}^d$ to $X^*$: $$Proj_{X^*}(x) = \arg\min_{x^* \in X^*} d(x, x^*)$$ where ...
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Intersection of compact sets in different spaces

Let $A$ be a compact set in $L^1$ and $B$ a compact set in $L^2$. Determine if $A \cap B$ is compact in $L^1$ or $L^2$ or both. My idea: Since $L^2 \subset L^1$ and $\Vert \cdot \Vert_2$ is stronger ...
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The smallest compact set that contains a bounded set in a metric space

I came across the following assertion: Let $X$ be a metric space. Let $Y\subset X$ be a bounded set. Then, there exists a smallest compact set $K$ in $X$ such that $K\supset Y$. How can I prove this? ...
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4 votes
1 answer
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Is the interval $[a,b]$ in $\Bbb{R}$ compact with the discrete metric?

I'm wondering whether or not a closed, bounded interval $[a,b]$ in $\Bbb{R}$ is compact with the discrete metric? I tried to show it wasn't by taking the set of singletons in $[a,b]$ as an open cover,...
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$\bar{B}_{1}(0) \subset \ell^{\infty}$ is not compact. [duplicate]

Show that: $\bar{B}_{1}(0) = \{x \in X | d(x,0) \leq 1 \} \subset \left\{\left(x_{k}\right)_{k \in \mathbb{N}} \subset \mathbb{R}^{n} \mid\left\|\left(x_{k}\right)_{k \in \mathbb{N}}\right\|_{\infty}&...
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1 vote
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An attempt at proving Dini's theorem.

By Dini's theorem, I mean $E$ is a compact metric space. $f: E \to \mathbb{R}$ and $f_n: E \to \mathbb{R} $ are all continuous functions. Moreover $f_n \to f$ pointwise. Suppose for each $p \in E$ ...
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Intersection of uncountable monotone decreasing compact sets

Let $C$ be a nonempty compact set. $A$ is a family of subsets of $C$, and $A$ is a totally ordered set with respect to the inclusion $\subset$. If $A$ is countable, then the intersection of all the ...
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2 votes
1 answer
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Compactly supported cohomology of open disk plus a point

Let $X=\{v\in\Bbb R^2:\|v\|<1\}\cup\{(-1,0)\}$, that is, the open unit disk union a point on its boundary. (Here, $(-1,0)$ refers to a point in the plane, not an interval.) What is $H_c^•(X)$, that ...
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5 votes
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Proving $f_n \rightrightarrows f$, provided that $f_n \rightarrow f$, each $f_n$ is increasing, and $f$ is continuous.

Clarification: this is not Dini's theorem. The title is highly succinct but here is the task in full detail: Let $[a,b]\in \mathbb{R}$ where $a<b$ are reals. Each $f_n: [a,b] \to \mathbb{R}$ is an ...
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4 votes
1 answer
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What does it mean to "augment" a set?

I am reading a book Hausdorff Compactifications and I don´t understand the sentence in bold. It is part of proof of the Theorem above (claim 1). I dont undersand in particular, what is meant by "...
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Reverse direction of Prokhorov theorem

I'm trying to prove the reverse of Prokhorov theorem. Could you verify if my attempt is fine? Let $(X, d)$ be a metric space and $\mathcal{P}(X)$ the set all Borel probability measures on $X$. Let $...
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2 votes
2 answers
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Is the Alexandroff one-point extension metrizable? [closed]

Let $(E,\tau)$ be a topological space and $\Delta\not\in E$; $E^\ast:=E\cup\{\Delta\}$ and $$\tau^\ast:=\tau\cup\underbrace{\left\{E^\ast\setminus B:B\subseteq E\text{ is }\tau\text{-closed and }\tau\...
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Showing that every convergent sequence in $\beta\mathbb{N}$ is eventually constant

I'm trying to show that every convergent sequence in $\beta\mathbb{N}$ is eventually constant. My professor told me to prove and use the following fact: Let $X$ be a Hausdorff space and let $(a_n)_n\...
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1 answer
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Show that there exists an infimum for this set of functions

Let $S\subseteq\mathbb{R}^n$ be a non-empty set. Consider that the set $C_S=\{\gamma:[0,1]\rightarrow\mathbb{R}^n|\gamma\text{ is continuous }, S\subseteq\gamma([0,1]), \gamma \text{ is Lipschitz}\}$ ...
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1 vote
2 answers
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Let $(X, d)$ be a separable metric space. There exists a compact metric space $(Y, d')$ and a map $T:X\to Y$ such that $X$ is homeomorphic to $T(X)$

In proving the reverse direction of Prokhorov theorem, I have to prove this auxiliary result. Could you verify if my attempt is fine? Let $(X, d)$ be a separable metric space. There exists a compact ...
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Does any weak form of compactness can force the conclusion of the closed graph theorem to be true?

Closed Graph Theorem (Topological Version , necessary condition ) : $X,Y$ be two topological space where $ Y $ is a compact Hausdorff space and $f:X\to Y$ be a map with $G_f=\{(x,f(x)):x\in X\}\subset ...
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Is the Alexandroff one-point extension always compact?

Let $(E,\tau)$ be a topological space and $\Delta\not\in E$; $E_\Delta:=E\cup\{\Delta\}$ and $$\tau_\Delta:=\tau\cup\underbrace{\{E_\Delta\setminus B:B\subseteq E\text{ is closed and compact}\}}_{=:\:...
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1 vote
1 answer
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Equicontinuity, compactness and uniform convergence.

Let $X$ be a compact topological space and let $K\subset \mathbb{R}^n$ be compact. Let $F = \{f_n\}_{n \ge 1}$ be a sequence of continuous functions where $f_n : X \to K$ that converges pointwise to $...
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1 vote
2 answers
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Exercise 4, Section 30 of Munkres’ Topology

Show that every compact metrizable space $X$ has a countable basis. [Hint: Let $\mathscr{A}_n$ be a finite covering of $X$ by $1/n$-balls.] My attempt: Approach(1): $B_n =\{ B(x, \frac{1}{n})| x\in X\...
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1 vote
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Metrizable Countable Compactness implies Compactness

I've read the other posts regarding this question but they all use obscure lemmas and other equivalences of compactness to prove their point. I want to start with countable compactness and ...
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2 votes
0 answers
68 views

Exercise 14, Section 30 of Munkres’ Topology

Show that if $X$ is Lindelof and $Y$ is compact, then then $X \times Y$ is Lindelof. My attempt: let $U=\{ U_\alpha \in \mathcal{T}_p | \alpha \in J\}$ be an open cover of $X \times Y$. Given $x_0 \...
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1 vote
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Are compositions of functions from compact function spaces compact?

Let $X\subset \mathbb{R}^d$ and $A\subset \mathbb{R}$ be a compact sets, where $C_B(X)$ and $C_B(A)$ denote the space of continuous and bounded real-valued functions $X\rightarrow\mathbb{R}$ and $A\...
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1 vote
1 answer
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about compact metric space and open set.

let $X$ be a compact metric space and $\{x_1,x_2,x_3,...,x_n\}\subset X$. Now let $$ A=\left\{a>0\,:\, \bigcup_{i=1}^{n} B_{a}\left(x_{i}\right)=X\right\}. $$ Prove or disprove : $A$ is open set ...
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Compact Riemann surface is sequentially compact.

Now, I try to prove that; M:a compact Riemann surface. $\forall \{P_j\}_{j\in N}\subset M$ (sequence of points) $\exists\{P_{j_k}\} _{k\in N}$ (subsequence of $\{P_j\}$) s.t. the subsequence converge....
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1 vote
1 answer
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Minimum distance from a point to a closed set in $\mathbb{R}^n$.

The task: S is a non-empty closed subset of $\mathbb{R}^n$ equipped with the Euclidean metric. Take $a \not \in S$. Show $\min \{ d(x,a) \ | \ x \in S \}$ exists. Below is my attempt. I wanted to ...
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2 votes
1 answer
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Prove or Disprove Compactness

Prove or Disprove: Suppose that $f$ is a real-valued function that is continuous on a nonempty set $S$ in $\mathbb{R}^n$ and that $f(S)$ is compact in $\mathbb{R}$. Then $S$ is a compact set $\mathbb{...
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1 vote
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Compact Subsets Of Bounded Completely Monotonic Functions

A smooth function $f$ : $(0,\infty) \to \mathbb{R}$ is said to be completely monotonic, if $(-1)^nf^{(n)} \geq 0 $ $\forall n$. Let BCM denote the set of all bounded completely monotonic functions on $...
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