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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86 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
119 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
36 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
42 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
60 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
69 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
69 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
65 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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186 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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0answers
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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35 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
32 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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0answers
19 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
176 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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0answers
16 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
85 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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1answer
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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2answers
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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2answers
85 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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39 views

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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2answers
47 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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58 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. ...
3
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1answer
112 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
76 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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0answers
81 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
40 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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1answer
39 views

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
55 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
30 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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0answers
81 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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2answers
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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0answers
46 views

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
73 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
49 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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191 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.
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2answers
87 views

Baire sets in locally compact Hausdorff spaces

(This is a follow-up to Compact $G_\delta$ subsets of locally compact Hausdorff spaces.) Suppose $X$ is a locally compact Hausdorff space. The Baire sets in $X$, denoted by $\mathcal Ba(X)$, comprise ...
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0answers
70 views

Compact $G_\delta$ subsets of locally compact Hausdorff spaces

Suppose $X$ is a locally compact Hausdorff space and $F$ is a closed subset thereof. Then of course $F$ is also locally compact and Hausdorff. Let $K$ be a subset of $F$, and suppose that $K$ is a ...
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1answer
73 views

A direct proof that a compact metric space is sequentially compact

I am looking for a direct proof (not by contradiction) that a compact metric space is sequentially compact, ie constructing a converging subsequence from any sequence. Thanks
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1answer
29 views

Are pseudocompact metric spaces complete?

Is there a way to show that pseudocompactness on a metric space implies completeness directly (without using sequential compactness)?
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1answer
49 views

Compact sets closed in Hausdorff spaces without choice?

An elementary proof that compact sets are closed in Hausdorff spaces involves making arbitrary choices based on the Hausdorff property. Is there a way to avoid invoking choice?
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175 views

If (X,d) is a separable metric space then there exists a metric d′ that is topologically equivalent to d and such that (X,d′) is totally bounded.

I know that this question Separability, total boundness and topological equivalence of metrics has been asked, but the only solution given is not valid. There is something I already knew: (Y, d2) ...
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2answers
27 views

Compact zero-dimensional $T_2$-topologies on $\mathbb{N}$

Let $\tau$ be a compact topology on $\mathbb{N}$ such that for every two points $m\neq n\in \mathbb{N}$ there is a clopen set $U$ containing $m$ but not $n$. Is $(\mathbb{N},\tau)$ isomorphic to ...
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1answer
196 views

Separability, total boundness and topological equivalence of metrics

The problem is: If $(X,d)$ is a separable metric space then there exists a metric $d'$ that is topologically equivalent to $d$ and such that $(X,d')$ is totally bounded. I know that if ...
4
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1answer
59 views

One point compactification of $[0,1]\times [0,1)$

What is one point compactification of $[0,1]\times [0,1)$? If we draw the figure we see that top line is missing and we've to add just one point to make it compact. So I think triangle will be ...