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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Compactness in a vector space

If $E$ is a normed space and $F$ is a subspace of $E$, how to prove that if $F\neq\{0\}$ then $F$ is not compact? I begin by this let $x\in F$ then $F=\bigcup_{x\in F} B(x,\varepsilon)$ how to say ...
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50 views

If $X$ is a non-compact metric space, can $X^n$ ever be compact?

Do there exist metric spaces $X$ such that $X^n$ is compact even though $X$ is not? Since compact spaces can have non-compact subspaces, e.g. $[0,1)\subset[0,1].$
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49 views

Compactification, definite function

Let $\hat{X}$ be the compactification of a Locally compact Hausdorff-space $X$. Show, that it exists an unique, continuous function $p_{\hat{X}}:\hat{X}\to X^+$, whose restriction on $X$ is the ...
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2answers
29 views

compactness of a set of sequences

I'm sorry if this is probably a stupid and not well-posed question but I'm really new to topology. I have two compact sets $U\subset \mathbb R ^n$ and $Y\subset \mathbb R$. Then I define $Q$ to be the ...
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3answers
68 views

Hausdorff of $X$ implies Hausdorff of $Y$ under some strange condition

Let $p:X\to Y$ be continuous surjective closed mapping s.t. $p^{-1}(y)$ is compact $\forall y\in Y$, prove that: (a) If $X$ is Hausdorff, then $Y$ is Hausdorff (b) If $Y$ is compact, then $X$ is ...
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Composition Series Analogous to Compactness?

The wikipedia page for group with operators makes the following claim about composition series as being analogous to compactness: The Jordan–Hölder theorem also holds in the context of operator ...
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1answer
44 views

Proof or definition of compactness in lecture notes?

I am baffled with what I am seeing. First, here's what is noted as a definition in my notes Let $X$ be a set and $A \subseteq X$. A cover of $A$ by subsets of $X$ is a family $(W_i)_{i \in I}$ of ...
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2answers
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Compactness theorem, propositional calculus

Please help me with this problem. Prove that if $\land \Phi \models \lor \Psi$ (both $\Phi$ and $\Psi$ infinite) then there exist $\phi_1,...,\phi_n$ from $\Phi$ and $\psi_1,...,\psi_m$ from $\Psi$ ...
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117 views

Characterization of Compact Space via Continuous Function

Let $(X,\mathfrak{T})$ be a topological space. We know that if $X$ is compact and $f:X\to \mathbb{R}$ be any continuous function then $f(X)$ is bounded since the continuous image of a compact set is ...
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2answers
28 views

Prove all closed subspace of a compact space are compact: Redundancy?

I see a redundancy in the following proof of the statement. First, we have a lemma that this proof uses A subspace $A \subseteq X$ is compact if and only if every open cover of $A$ by open subsets ...
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1answer
37 views

Three notions of compactness: minimal conditions for equivalence?

There exist 3 notions of compactness: $X$ is compact if any open cover admits a finite subcover; $X$ is sequentially compact if any sequence in $X$ has a convergent subsequence; $X$ is limit point ...
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17 views

Partial converses to extreme value theorem

Under what conditions can we establish a converse to the extreme value theorem? That is, for what topological spaces $(X, \tau)$ can we say that if $(\forall f \in C(X))(\exists c \in E) \left( f(c) = ...
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1answer
15 views

Problem to demonstrate that it is compact

Let $X$ be a Normed Vector Space. For any $x\in X$ and $r>0$, let $W:=\{y∈X:∥y−x∥≤r\}$. Prove: $W$ is closed and if $\dim(X)<\infty$ $W$ is compact. I have no problems show that it is closed, ...
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44 views

Metric space on $\mathbb{R^n}$ where Heine-Borel criterion does not hold

Heine-Borel criterion of $\mathbb{R^n}$ : closed and bounded $\implies$ compactness Give an example of a metric space in $\mathbb{R^n}$ where this criterion does not characterize compactness ...
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0answers
56 views

How to prove a set is closed

Let $\left(\Omega, \mathcal{F}, \mathbb{P}\right)$ be a finite probability space equipped with a filtration, i.e an increasing sequence of $\sigma$-algebras included in $\mathcal{F}$ : $\mathcal{F}_0, ...
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1answer
35 views

Compactness and sequences in $\mathbb{R}^{n}$

Why is it that: If $A$ is a compact set and $\left ( a_{n} \right )$ a sequence in $A$, then there is a subsequence $\{a_{n_k}\}$ such that $\lim_{k\to\infty} a_{n_k}=a$ with $a\in A$. I get ...
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25 views

Prove the boundary is a compact 1 manifold

A closed surface with boundary is a compact connected topological space $B$ with the property that each point $p \in B$ has an open neighborhood $U$ homeomorphic to either: $\{(x, y) \in \mathbb{R^2}|...
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Is it possible to construct Hausdorff compact topology on every set?

I'd like to know if it's possible to construct Hausdorff compact topology on every set. Assume the axiom of choice if needed. Thanks for ideas.
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20 views

Sum of compact sets is compact without using continuity [duplicate]

I want to show that if $A$ and $B$ are compact sets, then $A+B$ (that is, the set $\{a+b : a \in A , b \in B\}$) is compact. I know that $A+B$ is bounded, but am having trouble showing that it is ...
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Compactness, why is $(0,1)$ not compact? I need the “thought process” [duplicate]

See, I am told that $(0,1)$ is not compact as a subspace of $\mathbb{R}$. Question is, how do I conclude that? The hint says $(\epsilon,1)_{\epsilon>0}$ does not have a finite subcover. But ...
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2answers
37 views

Function is continuous if graph is compact.

Let $X$ be a Hausdorff space and let $f:X\to \mathbb{R}$. If grapph of $f$ is compact we have to show that $f$ is continuous. Since every closed subset of a Hausdorff space is closed, therefore ...
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1answer
51 views

Property of compact metric space

Let $X$ be a compact metric space. Let $A$ be a closed subset of $X$ and let $x\in X$ be a point not in $A$. Show that there exists two disjoint open sets $U$ and $V$ such that one contains $A$ and ...
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Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
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1answer
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Continuous function on a non-compact set

I'm trying to show if $X$ is non compact ($X \subseteq \mathbb{R}$) then there is a cont function $f:X \rightarrow \mathbb{R}$ which is bounded but doesn't attain it's bounds. I'm trying it for a set ...
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86 views

Why can't a open interval in $\mathbb{R}$ be compact?

If I choose $(0,1)$ and do a covering like this: $(-1,1/2)$ and $(1/3,2)$ it's a finite covering, then $(0,1)$ is a compact set?
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63 views

Why does this proof work: Closed unit ball in $C_0$ is not compact

I know that this question has been asked to death, and multiple solutions are given, but I still don't understand why the "standard" proof works Following Show that the closed unit ball $B[0,1]$ in $...
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1answer
50 views

A question about closed (but not necessarily compact) connected subsets of Euclidean spaces.

Is the following statement true?...... If $C$ is a non-degenerate closed and connected subset of the Euclidean plane $\mathbb R ^2$ and $p$ is any point of $C$, then there exists a connected ...
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64 views

Sequential compactness implies compactness: what is wrong with this argument?

Definitions A filter is a poset $(I,\leq)$ such that for any $\alpha,\beta\in I$ there is $\gamma\in I$ such that $\gamma\geq\beta,\gamma\geq\alpha$. A net in a set $X$ is a function from a poset to ...
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1answer
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Problem following proof of Šmulian theorem for separable space

I tried to solve Problem 10 on p. 464 of Brezis to get a proof of part of the Eberlein-Šmulian theorem, precisely the equivalence between compactness and sequential compactness in the weak topology of ...
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About accumulation point in compact metric space

Let $(X, d)$ be a compact metric space, and $\{x_n\}_{n\in N}$, $\{y_{n,m}\}_{m,n\in N}$ be subsets of $X$. Question: Is there a subsequence $\{n_k\}$ such that $x_{n_k}\rightarrow x$ and $y_{n_k,m}...
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E is infinite subset of compact set, then is E' also a subset?

Here's a theorem in Rudin's Principles of Mathematical Analysis. 2.37 Theorem: If E is an infinite subset of a compact set K, then E has a limit point in K. Proof: If no point of K were a limit ...
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1answer
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Bounded sequence in $W^{1,p}$ converging to a non-differentiable function in $L^p$

Let $U = B(0,1)$ be the unit ball in $\mathbb R^n$, $p>1$ and $\{u_k \}$ a bounded sequence in $W^{1.p}(U)$. The Rellich-Kondrachov compactness theorem tells us that there is a subsequence $\{ u_{...
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1answer
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Normalized measure over compact metric spaces

Consider the following definitions. Let $M = (V,T,d)$ be a compact metric space with finite diameter $$D = D(M) = \max d(x,y), ( x, y \in M)$$ and a finite normalized measure $\mu$$M$(.), ($\mu(.)$...
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1answer
25 views

Why is the the following statement not equivalent to compactness?

Well it comes down to word-play again. I'm confused to the core of my bones as to why the following isn't equivalent to saying that a space is compact Every open cover is finite. A compact set ...
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How to define $[-\infty, \infty]$ or $[0, \infty]$?

I am familiar with basic undergraduate topology. For example, I know the process of one point compactification of a non-compact topological space, and how it applies to, say, $\mathbb R^2$. My ...
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1answer
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Discuss about compactness of these sets

My question is: How can I see if (in $\mathcal H=\mathcal l (\mathbb{N} )$ $B_1=\left\{ u | \frac{|u_k|}{k^2}\leq1 \right \}$ ,$B_2=\left\{ u | \frac{|u_k|}{log(1+k)}\leq1 \right \}$ are compacts or ...
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1answer
28 views

Proof of being a compact set [closed]

I'm trying to solve this problem but I'm really stuck and it would be nice if someone can explain me proof or any hint for this problem. Let $X \subset\mathbb R^N$ be a nonempty compact set, and $f: ...
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Why is $ \{(1/2)^n : n \in \mathbb{N} \} \cup \{ 0 \} $ not compact?

$ S = \{(1/2)^n : n \in \mathbb{N} \} \cup \{ 0 \} $ is obviously bounded and infinite. It also looks totally disconnected to me (it is not and does not contain as its subset an interval with more ...
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1answer
40 views

Does compact set have always content

A set is said to have content iff it's boundary have content zero. So does compact set have always content? I can't really find a way to proof this, but for all examples that I can think of (...
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Which of the following are compact I need Hint…

Which of the following are compact? $\{(x,y) \in \mathbb{R}^2 :(x-1)^2+(y-2)^2=9\} \cup \{(x,y) \in \mathbb{R}^2: y=3\}$. 2.$\{(\frac{1}{m},\frac{1}{n}) \in \mathbb{R}^2:m,n\in \mathbb{Z}-\{0\}\} \...
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What is the $\epsilon$ neighborhood of a subset in $\mathbb R^2$ in $\mathbb R^n$

Let X denote the subset $(-1,1) \times 0$ of $\mathbb R^2$ and let U be the open ball B(0,1) in $\mathbb R^2$ which contains X. Show that there is no $\epsilon > 0$ s.t the $\epsilon$-neighborhood ...
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How can I show $R^n$ is dense in $S^n$?

How can I show $R^n$ is dense in $S^n$? I wanted to show $S^n$ is compactification of $R^n$. for this I need $R^n$ is not compact, for this there is no problem, and $S^n$ is compact, I did it with ...
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1answer
39 views

With justification, determine whether or not the following space is compact.

The space in question is the Hausdorff topological space with base β: β = {U(a, b) : a, b ∈ Z, b > 0}, where U(a, b) = {a + kb : k ∈ Z} . (I have confirmed that this in fact a base of a Hausdorff ...
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1answer
53 views

Show that a sequence of uniformly bounded continuous functions with Lipschitz condition is pre-compact in the space of bounded continuous functions

I am attempting to solve the following problem: Let the sequence of continuous functions $\{x_{n}(t) \}_{n=1}^{\infty}$, $0 \leq t < \infty$ be uniformly bounded on $t \in [0, \infty)$ and on ...
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1answer
71 views

Why must the countable sets shown in this example be closed?

In the book Introductory Real Analysis (Kolmogorov & Fomin, English Translation by Richard Silverman Pg 95, see below) It says: "the sets $X_n= \{x_n, x_{n+1}, ...\}$ form a centered system of ...
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Show that $ \operatorname{Sp}(n)=\{A \in M_n(\mathbb{H}) \mid AA^*=I=A^*A\} $ is a compact group

Let $M_n(\mathbb{H})$ be the set of all $n \times n$ matrices with entries in the quaternions $\mathbb{H}$. For $A=(a_{ij} ) $ let $ A^*=(a^*_{ij} ) $ be the matrix with $a^*_{ij}=\bar{a}_{ij} $, ...
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Question about product of compact spaces being compact

I'm reading through a proof that for $X,Y$ compact, $X\times Y$ compact with the standard topology. The proof begins by saying: consider a specific type of cover $\mathcal{C}$ where each $T\subseteq \...
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Diagonal argument involving relative compactness of densities

There is a claim from a paper which I do not understand: Let $D$ be a domain in $\mathbb{R}^d$. Let $(p^{\eta})_{\eta >0}$ be a family of densities for random variables on $(C[0,T], \mathbb{R}^...
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1answer
23 views

How to index compact set problem?

For any α∈I, if Aα is compact set then ∩(α∈I)Aα is compact set. Tomorrow, I will mid-exam, i'm very dizzy. Please, prove that problem.
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Closed graph of a function

I have difficulties to answer at that question: Let $X$ be a Hausdorff and compact topological space, and let $Y$ be a topological space. Let $f:X→Y$ be such that $G(f) = \{(x,f(x))|x∈X\} $ is a ...