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, sequential spaces

A compact, Hausdorff space $X$ is sequential if each for each $A\subset X$ and $x\in \overline{A}$, there exists a countable set $A_0\subset A$ such that $x\in \overline{A}_0$. I am asked to show ...
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54 views

Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
4
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1answer
72 views

Locally Compact Spaces: Characterizations

For Hausdorff spaces the following are equivalent: Every point admits a compact local base. Every point admits a compact neighborhood. Every point admits a precompact neighborhood. Every point ...
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1answer
32 views

Bound for Integrator Operator

Let $E = L^p(0,1)$ with $1 ≤ p < ∞$. Given $u ∈ E$, set $$Tu(x):=\int_0^x u(t)dt$$ Prove that $T$ is compact on $E$. I would like to use Ascoli-Arzela', but I need to prove: $$|T u(x) − T u(y)| ...
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1answer
25 views

Checking that $C_{0}(X)$ is a vector space

I am trying to prove that $C_0(X)$ is closed subspace of $C_b(X)$ (bounded continuous functions) Given, $X$ is locally compact. $C_0(X)$ is the space of all continuous functions $f:X \to F$ (field of ...
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1answer
46 views

Check if $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ is compact

I want to check, if this set is compact: $M = \{z \in \mathbb{C}| z = \frac {1}{n} + \frac {i}{m} \ with \ \ m,n \in \mathbb{Z} \backslash \{ 0 \} \} $ Thoughts: $z:= a +bi$ real part $a$ is ...
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1answer
29 views

What's the meaning of the state space with locally compact topological space?

I have encountered a statement in one paper describing the continuous-time controlled Markov chain with space state which is locally compact topological space. What does this mean? In my previous ...
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2answers
29 views

Condition that a local homeomorphism be a covering map.

Let be $f:Y\to X$ a local homeomorphism, with $Y$ a compact space and $X$ a Hausdorff connected space. How can I show that, for each $x\in X$, $p^{-1}(x)\subset Y$ is finite? So, is clear that $f$ is ...
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31 views

Compact subset of space of matrices and compactness verification of a set of eigenvalues

Let $M_n(\mathbb R)$ be the vector space of real matrices of size $n$ , identified with $\mathbb R^{n^2}$ ; let $X \subseteq M_n( \mathbb R)$ be a compact set ; let $S \subseteq \mathbb C$ be the set ...
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1answer
111 views

Countable union of compact sets is compact?

Let $A_0$ be a compact set (closed and totally bounded in some metric space) and consider a sequence of sets $A_n=\{x:d(x,A_0)<1/n\}$. For each $n$, $A_0\subset B_n\subset A_n$ is compact. ...
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1answer
142 views

Uniform Convergence to the Exponential Function over a Compact Interval

I'm trying to show that the sequence of functions $f_n(x)=(1+(x/n))^n$ converges uniformly to $f(x)=e^x$ over any compact interval of the real line. We're assuming that it converges pointwise. Here is ...
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1answer
43 views

James' theorem—going from the separable case to the general case

Consider the following famous theorem by Robert C. James (1964): Let $X$ be a Banach space over $\mathbb R$ and $C$ a non-empty, bounded, weakly closed subset. Then, $C$ is weakly compact if and ...
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1answer
43 views

Metric $p := p(x,y)= \min(|x-y|, 1- |x-y|)$ $x,y \in [0,1)^2$. Prove metric space is compact.

Help! I know that $X$ is Compact if every sequence in $X$ has a subsequence converging to a point in $X$. Also we have that $X$ is a bounded infinite subset in the real numbers. I think it's quite ...
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1answer
64 views

Complement of a point of a Compact Connected Hausdorff Space has no compact maximal connected subspace

This question is a slight modified version of Compact Connected Hausdorff Space has no compact component in the complement of a point Let $X$ be a Hausdorff Compact Connected Space. Prove that ...
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1answer
76 views

Compact Connected Hausdorff Space has no compact component in the complement of a point

Let $X$ be a Hausdorff Compact Connected Space. Prove that $X\setminus\{x\}$ can't be expressed by the disjoint union of two connected sets with one them being compact.(lets assume the empty set is ...
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1answer
23 views

Show compactness of $E\cup S_1$

Consider $$ S_1:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\},\\E:=\left\{0\right\}\cup\bigcup_{n\in\mathbb{N}}\left\{(1-2^{-n})e^{\pi i k/2^n}: ...
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1answer
71 views

Check if the given set is Connected and Compact.

$S=\left\{\dfrac{x^{2}}{1+x^{2}}:x \in \mathbb R\right\}$ Since $S$ is not closed (the limit point $1$ does not belong to the set), so I concluded that $S$ is not compact. I am confused about ...
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2answers
71 views

Regularity of Dirac measure on Baire sets

Suppose $X$ is a locally compact Hausdorff space. Define the Baire sets in $X$, denoted by $\mathcal Ba(X)$, to be the smallest $\sigma$-algebra that contains all compact $G_\delta$ subsets of $X$. ...
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191 views

non-symmetric version of compact = totally bounded + complete

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is ...
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35 views

How do we call such a compactification?

Let $E$ be a denumerable set and let $\mathcal{F}$ be a collection of bounded functions. In the reading we had a compactification of $E$ with respect to $\mathcal{F}$, denoted by ...
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36 views

Open sets in locally compact spaces [closed]

Is every open set in a locally compact space an $F_\sigma$ set? Not assuming any separation axioms
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1answer
121 views

Show that $\hat{E}\setminus E$ is homeomorphic to $S^1$

Set $\mathbb{D}=\left\{z\in\mathbb{C}: \lvert z\rvert <1\right\}$ and define $\mathcal{P}\colon\mathbb{D}\times\mathbb{D}\to\mathbb{R}$ by $$ \mathcal{P}(x,y):=\begin{cases}\frac{1-\lvert ...
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3answers
37 views

Can someone formalize 'compactness of set' defined as follows?

My text book says A set K ⊆ R is compact if every sequence in K has a subsequence that converges to a limit that is also in K. Wondering how to formalize this statement, my trial was this. ...
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1answer
34 views

Compactness is independent of the ambient space

In Rudin's PMA book, 2.33 Theorem states that: Suppose $K \subset Y \subset X$. Then $K$ is compact relative to $X$ if and only if $K$ is compact relative to $Y$. I cannot write down the proof as I ...
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20 views

Proving compactness in a geometric scenario

Let $C$ be a compact subset of $R^2$. Let $D$ be the set of all pairs of points $(P,Q)$ from $C$, such that the open segment between $P$ and $Q$ is contained in $C$: $$D = \{(P,Q)|P\in C, Q\in C, ...
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2answers
186 views

Continuous function on a compact set with no fixed points

I'm reviewing this problem for my analysis qual. Let $f:X\rightarrow X$ be a continuous mapping from a metric space to itself. Assume $f $ has no fixed points. Prove that, if $X $ is compact, ...
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18 views

Compactification: density of a uniform space $X$ in the spectrum of $UC^b(X)$

First, a small motivation: Suppose we are looking for a compactification of uniform spaces, satisfying an universal property similar to the one of the Stone-Čech compactification of a locally compact ...
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1answer
90 views

Question about quotient of a compact Hausdorff space

I am reading the book 'Algebraic Topology' by Tammo Tom Dieck. On page 12 in the proposition 1.4.4 he states that : Let $X$ be a compact Hausdorff space and $f : X \rightarrow Y$ be a ...
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42 views

Topologial properties of $f([1,3]^3)$ where $f(x,y,z)=x^2+2xz+y$

If $f(x,y,z)=x^2+2xz+y$, determine $f([1,3]^3)$ and characterize this set in terms of openness, closedness, completeness, compactness and connectedness. Since $[1,3]^3$ is compact then ...
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40 views

Conditions that guarantee the existence of a largest piece

Let $m$ be the area measure on $R^2$. Let $S$ be a nonempty set of measureable subsets of $R^2$ ("pieces"). Define the largest piece in $S$ as: $$\arg \max_{s\in S} m(s)$$ I am looking for ...
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95 views

What is the one point compactification of the reals?

In several of my questions this theorem has come up. What is the one-point compactification of the reals? Does it have to do with limits and dividing by $0$? I vaguely remember something about a ...
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Continuity of two variable function with compactness and supremum norm

Please assist me with the following homework problem: Let $X$ and $Y$ be metric spaces and suppose that $Y$ is compact. Let moreover $f: X \times Y \to R$ be a continuous function, and define ...
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1answer
28 views

Question about Compactly Supported functions

Suppose I have a compactly supported function $f$ defined with $supp f \subset I= ]0,1[$. Let $K=supp f$. Is this statement true: We can assume without loss of generality that there exists $ ...
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49 views

If $f$ is a continuous function from $R^3$ to $R$ and $K⊂R^3$ is compact, show that there exist two points $a, b ∈ K$ so that $f(K)⊂[f(a),f(b)]$

If $f$ is a continuous function from $R^3$ to $R$ and $K⊂R^3$ is compact, show that there exist two points $a, b ∈ K$ so that $f(K)⊂[f(a),f(b)]$. When is $f(K)=[f(a),f(b)]$? What I believe is the ...
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59 views

If $T \models \phi$ then there is a finite subtheory $T' \subset T$ such that $T' \models \phi$

Use the Compactness Theorem to show: if $T \models \varphi$ then there is a finite subtheory $T' \subset T$ such that $T' \models \varphi$. I don't see how I can use the compactness theorem here. ...
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1answer
113 views

Why compact-open topology implies joint continuity?

On page 76, A guide to topology by Steven Krantz, there is a motivating question: If $\mathscr E$ is a family of function from $S$ to $\mathbb R$, then under what circumstances is the mapping ...
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1answer
80 views

Compactness and completeness in groups

I know that, in metric spaces, compactness implies completeness. In fact, (i) compactness is equivalent to the fact (ii) every infinite set has an accumulation point and to the fact that (iii) any ...
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105 views

Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

First of all sorry for my bad English, I'm an Italian student, hope to let you understand! I'm having a little troubles with the proof of the Riesz-Schauder theorem for Compact Operators. Some infos ...
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1answer
41 views

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?

If $X$ is a compact metric space and $f:X \to Y$ is a continuous map , where $Y$ is another metric space , then is $f(X)$ a complete subset of $Y$ ?
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Looking for counter example - compactness theorem

Let $S$ be a family of sets. We say a subset $S'\subseteq S$ is good if we can choose from every set $A\in S'$ a representative $x_A$ s.t.: For every three sets $A,B,C\in S'$ it holds that $(x_A + ...
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1answer
56 views

Continuous extension on compact set in $\mathbb{R}^n$

I'm an undergrad student reading through Deimling's Nonlinear Functional Analysis and have come across the following proposition. Let $A\subset\mathbb{R}^n$ be compact and $f:A\to\mathbb{R}^n$ be a ...
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1answer
31 views

Is this a compact metric space?

Consider a fixed set of finite discrete symbols $\mathcal{A}$. Equip $\mathcal{A}$ wit the discrete topology which we denote by $\theta$, and $\mathcal{A}^{\mathbb{Z}^d}$ with the product topology, ...
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82 views

Approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets?

Are there some common ways to approximate the unit ball in an infinite-dimensional Hilbert space, by compact sets? (note that the unit ball isn't compact.) My goal is to prove a statement which holds ...
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2answers
58 views

All neighborhoods of a compact subset of an open space are subsets of that open space

Let $K$ be a subset of $U$, with $K$ compact and $U$ open. Prove that there is an $\epsilon > 0$ such that for all $p$ in $K$, a neighborhood of radius $\epsilon$ of $p$ is a subset of $U$. Note: ...
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43 views

Intersection of compact convexes

Let $C_1,C_2,C_3,C_4$ be compact convexes of $\mathbb{R}^2$ such that $C_1\cap C_2\cap C_3\neq\emptyset,C_1\cap C_2\cap C_4\neq\emptyset,C_1\cap C_3\cap C_4\neq\emptyset,C_2\cap C_3\cap ...
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Prove that the continuous $f: \mathbb C \to \mathbb R$ has a global max and min

I am having this continuous transformation $f: \mathbb C \to \mathbb R$ and $\ f\ (\mathbb C)$ is bounded Now I have to prove that there are a global maximum and a global minimum. My thoughts: I ...
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1answer
80 views

$\mathbb{N}$ is a Compact Space with the Co-finite Topology?

Let $X$ be the topological space on the set $\mathbb{N}$ with the cofinite topology. I am having a hard time seeing why this is compact in the topological sense. If each open $n$-hood on $X$ ...
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1answer
52 views

A compact set, which is not closed.

I'm looking for a compact set, which is not closed. I read somewhere that $Z^+$ are compact and not closed, but I don't understand why. Are there any other examples of compact sets that are not ...
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195 views

defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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1answer
71 views

If a set is closed, why is that set intersected with a compact set closed?

If $F$ is a closed subset of $K$ and $K$ is compact, why is $F \cap K$ closed? Progress I just realized compact subsets of a metric space are closed.