# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $T\subseteq \... 0answers 9 views ### Diagonal argument involving relative compactness of densities There is a claim from a paper which I do not understand: Let$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}^...
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 ...