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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3
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1answer
148 views

Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology?

My question is: Why is $[0,1]^\mathbb{N}$ not countably compact with the uniform topology? How do you prove this? Do you use the countable open covering or do you use the accumulation point ...
3
votes
1answer
167 views

Intuition behind compact subspaces of a metric space

I've read up on compactness in a metric space and have found a few definitions (let $X$ be a metric space and $E \subset X$ in all the following): $E$ is compact in $X$ if for every open covering of ...
3
votes
1answer
174 views

Stone-Čech compactification. A completely regular topological space is locally compact iff it is open in its Stone-Čech compactification.

I would like to show that a completely regular topological space is locally compact iff it is (weak-star) open in its Stone-Čech compactification. Does this hold in general? I.e given a compact ...
3
votes
1answer
75 views

How to show this space $X$ is countably compact, first countable?

Consider the subspace $X$ of $(2^\omega)^+$, i.e., the smallest cardinal greater then $2^\omega$, equipped with the ordered topology consisting of all ordinals of countable cofinality. How to ...
4
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1answer
71 views

Is a minimal Hausdorff uniformity compact?

Let $(X,\mathcal D)$ be a Hausdorff uniform space and for each Hausdorff uniformity $\mathcal U$ on $X$, $$\mathcal U \subseteq\mathcal D\to \mathcal U =\mathcal D$$ Is $(X,\mathcal D)$ compact?
0
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1answer
113 views

equivalence of compactness and countably compactness

Is there a way to prove that in metric spaces, compactness and countably compactness are equivalent, without using the Bolzano Weierstrass Property?
1
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1answer
19 views

compactness - show that there an absolute max

This is a practice question from "Advanced Calculus, Folland. Chapter 1.7 Q.4) Suppose $\quad S\subset { R }^{ n }\quad $ is compact $\quad f:S\longrightarrow R\quad $ is continuous and ...
2
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1answer
68 views

exercises in compactness

I am working on some practice problems on Compactness. (Q.1.a Chapter 1.7 in Advanced Calculus, Folland) The question is : Give an example of : a closed set $S\subset R\quad$ and a continuous ...
3
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1answer
43 views

No unbounded real continuous function on $X$ can be extended to a continuous real function on $\beta X$

By the Čech-Stone compactification theorem, I know that if $X$ is Tychonoff and $f:X\to [a,b]$ is continuous then $f$ can be extended to $\hat{f}:\beta X\to [a,b]$. How can we show that no unbounded ...
1
vote
1answer
48 views

Show $\tau=\tau^*$ if $\tau^*\subset \tau$ [duplicate]

Let $(X,\tau)$ be compact and $(X,\tau^*)$ be a Hausdorff space. How can we show that $\tau=\tau^*$ if $\tau^*\subset \tau$?
1
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1answer
223 views

SHOW that there are infinitely many equivalence classes of formulas

Let $\mathcal{Q}$ denote the additive group of rational numbers, i.e. the structure $\left<\mathbb{Q}; +; 0\right>$. Let $\mathcal{L}$ be the language of $\mathcal{Q}$ and let $T$ be the ...
7
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2answers
108 views

dense subspace of $\beta \Bbb N \times \beta \Bbb N$

Let $\beta \Bbb N$ be a Čech-Stone compactification of the discrete space $\Bbb N$ and fix a point $p\in \beta \Bbb N\setminus \Bbb N$. Put $X=\Bbb N\cup \{p\}$ and $Y=\beta \Bbb N \setminus \{p\}$. I ...
6
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4answers
2k views

A example of closed and bounded does not imply compactnesss in metric Space

Let $X$ be the integers with metric $ρ(m,n)=1$, except that $ρ(n,n)=0$. Check that $ρ$ is a metric. Show that $X$ is closed and bounded, but not compact. This is a "made-up" example demonstrating ...
4
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1answer
109 views

Question about Čech-Stone compactification

Let $\beta X$ be the Čech-Stone compactification of $X$ and $p\in \beta X\setminus X$. Is it true that $\{p\}$ can not be a $G_\delta$ set ?
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2answers
34 views

Prove that $D ⊂\Bbb R^{n}$ is compact iff whenever {$C_{α}$} is a collection of relatively closed subsets of $D$ with the property $∩ C_{α} = ∅$

Prove that $D ⊂\Bbb R^{n}$ is compact if and only if whenever {$C_{α}$} is a collection of relatively closed subsets of $D$ with the property $∩ C_{α} = ∅$ , there is a finite subcollection satisfying ...
2
votes
3answers
232 views

Norm equivalence of a vector norm and its induced matrix norm using compactness argument

I have a theorem in my book on matrix computations that states the following: A vector norm and its induced matrix norm satisfy the inequality: $\|Ax\|\leq \|A\|$$\|x\|$ where A $\in R^{nxn}$ and x ...
5
votes
1answer
548 views

closed bounded subset in metric space not compact

Let $\ell^{\infty}$ be the space of bounded sequences of real numbers, endowed with the norm $\|\mathbf x\|_\infty=\sup_{n\in N}|x_n|$, where $\mathbf x=(x_n)_{n\in\Bbb N}$. Prove that the closed ...
7
votes
1answer
125 views

Stone-Čech compactification of completely regular space

Suppose $X$ is a completely regular space. Let $M$ be the set of nonzero algebra homomorphisms from $BC(X,\mathbb{R})$ to $\mathbb{R}$, equipped with the topology of pointwise convergence. Show that ...
2
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0answers
125 views

For a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$, if $F$ is compact on compact subsets of $X$, then $F$ is compact on $X$

The problem as stated in the title isn't quite correct. Let $X$ be a topological space. What I have is a family of functions $F\subset C(X)$ in the metric space $(C(X),d)$ which on compact subsets ...
2
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1answer
352 views

Compactness, Local Compactness, Completeness

Clearly, every compact metric space is locally compact. I always get confused when completeness is introduced into the picture. Which of the following are true? What are some easy counterexamples to ...
6
votes
2answers
127 views

Does this characterize compactness?

Recall that a collection of sets $\mathcal{A}$ has the finite intersection property if for all finite $\mathcal{B} \subseteq \mathcal{A}$ it holds that $\bigcap \mathcal{B} \neq \emptyset$. In terms ...
1
vote
1answer
63 views

Compact metric space: proof $\text{diam}(K)$

I am to assume that $K$ is a compact metric space. I must prove that there are two points $x,y$ contained in $K$ such that $d(x,y)=\text{diam}(K)$. Recall $\text{diam}(K)= \sup \{ d(x,y) \mid x,y ...
5
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1answer
164 views

compact and locally Hausdorff, but not locally compact

I wonder if there is a compact and locally Hausdorff space $X$ which is not locally compact, in the sense that every point has a neighborhood base consisting of compact sets. A space is called ...
7
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1answer
89 views

A question concerning compactness - Topology

Prove that if $X$ is a compact space and $H = \{h_{\alpha} : \alpha \in A\}$ is any collection of closed subsets with the property that $\cap_{\alpha}h_{\alpha} = \emptyset$, then there is a finite ...
2
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3answers
2k views

Show that the infinite intersection of nested non-empty closed subsets of a compact space is not empty

I'm given the following problem: Suppose that for every $n\in \mathbb{N}$ $V_n$ is a non-empty, closed subset of a compact space $X$, with $V_n \supseteq V_{n+1}$. Now I have to show that ...
2
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4answers
186 views

Why is this family of open sets a cover for $(0,1)$?

I'm dealing with some beginner examples of compact spaces. The definition I am given is A subset $A$ of a topological space $X$ is compact if every open cover for $A$ has a finite subcover for ...
26
votes
1answer
461 views

When is Stone-Čech compactification the same as one-point compactification?

For the space $\omega_1$ (with the order topology) we have $\beta\omega_1=\omega_1+1$ (or $\beta[0,\omega_1)=[0,\omega_1]$, if you prefer this notation), i.e., it is an example of a space for which ...
5
votes
1answer
142 views

Every compact space is a continuous image of a compact Moscow space.

From WikiPedia, every compact metric space is a continuous image of the Cantor set. and in Topological Groups and Related Structures Ex 6.3.a. every compact space is a continuous image of a ...
0
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1answer
121 views

Countable product of finite sets with a new metric, compact?

Suppose we have a finite set $E$. Is it true that $E^n$ is compact? The metric on $E^n$ is : $$d(\omega,\omega\prime)=\begin{cases}2^{-\inf \{ n \in \mathbf N:\omega _n \ne \omega'_n\} }&{\omega ...
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4answers
51 views

Question about the definition of compact topological space

A topological space X is called compact if each of its open covers has a finite subcover. http://en.wikipedia.org/wiki/Compact_space The finite subcovers are also open covers of $X$ so ...
3
votes
1answer
114 views

Compactness for a set of functions.

Let $\Omega\subset\mathbb{R}^n$ be a open set and write $\Omega=\cup_{i=1}^\infty Q_i$, where $Q_i$ is a compact set. Define in $L_{loc}^p(\Omega)$ the metric ...
9
votes
1answer
183 views

Ideal in compact Hausdorff space

This is exercise 70, chapter 4. from Folland (page 142) Let $X$ be a compact Hausdorff space. An ideal in $C(X, \mathbb{R})$ is a subalgebra $J$ of $C(X, \mathbb{R})$ such that if $f\in J$ and $g\in ...
4
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0answers
58 views

A question on countably compact space under CH

The question is also posted here. A regular space $X$ is star compact (which implies pseudocompact) with $G_\delta$-diagonal star countable first countable $e(X)\le \aleph_0$ ( in fact it implies ...
4
votes
1answer
134 views

A question on countably compact

Here is a Lemma from this paper, which maybe helpful for the answering my question. A topological space $X$ is said to be star compact if whenever $\mathscr{U}$ is an open cover of $X$, there is a ...
5
votes
1answer
84 views

Artinian topological space are compact

Call a topological space $X$ Artinian if every nested sequence of closed sets $$C_1 \supset C_2 \supset C_3 \supset \cdots$$ is eventually constant. Prove that if $X$ is Artinian then it is ...
3
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2answers
151 views

Imposing the topology of open rays in $\Bbb R$

After having received Brian M. Scott's permission (see comments in the selected answer) I am integrating his suggestions with my own solutions to form a complete answer to the questions apperaing ...
3
votes
1answer
136 views

$W^{1,p}$ compact in $L^\infty$?

Is $W^{1,p}(0,1)$ compactly contained in $L^\infty(0,1)$? Can I use this to show that I can select a sequence $(u_{n_k})$ from every bounded sequence $(u_n)$ in $W^{1,p}(0,1)$ such that $\lVert ...
2
votes
1answer
302 views

Compactness in $L^2$

Let $n\in\mathbb{N},\ T>0$ and $$\Sigma:=\{\sigma\in L^{2}(0,T;\mathbb{R}^{n}):\sum_{i=1}^{n}\sigma_{i}(t)=1,\ \sigma_{i}(t)\geq 0\ \hbox{almost everywhere}\}.$$ Do we then have ...
5
votes
3answers
173 views

Compactness and Strictly Finer Topologies.

If $(A,\tau{_1})$ is a compact Hausdorff space and $\tau{_2}$ is a strictly finer topology on X, can $(A, \tau_{2})$ be compact?
32
votes
4answers
643 views

To show that the set point distant by 1 of a compact set has Lebesgue measure $0$

Could any one tell me how to solve this one? Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$ Show that $A$ has Lebesgue measure $0$. Thank you!
5
votes
1answer
148 views

Proving $\mu(A)=\inf\{\mu(O) \mid A\subseteq O, O \text{ open}\}$

Can someone please help me show, why in a compact metric space $(X,d)$ we have have $$ \mu(A)=\inf\{\mu(O) \mid A\subseteq O, O \text{ open}\}$$ and $$ \mu(A)=\sup\{\mu(K) \mid K\subseteq A, K \text{ ...
1
vote
2answers
47 views

Some compactness related property

Let $X$ be a topological space. It is known that $X$ is compact if and only if every family of closed sets satisfying the finite intersection property has nonempty intersection. Is there some kind of ...
1
vote
1answer
184 views

When Cantor's Intersection theorem wont work with closed sets

Give an example to show that Cantor's Intersection Theorem would not be true if compact sets were replaced by closed sets. Compact set is closed and bounded, so what Im going to find is something ...
3
votes
1answer
348 views

A locally compact subset of a locally compact Hausdorff space is locally closed

Let $X$ be a locally compact and Hausdorff space. Show that if $Y \subset X$ is locally compact, then $Y$ is locally closed, in essence $Y$ is an open subset of $\overline{Y}$, where $\overline{Y}$ ...
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votes
2answers
120 views

Are these sets compact? [closed]

I've some problems concerning this question: Are the following sets compact in $C_{[0, 1]}$ where $ d(x(t), y(t))=\sup_{[0,1]}|x(t)-y(t)|$: $${\{x(t) \mid x(t)=e^{t-a}, a>0\}},~~{ \{ x(t) ...
6
votes
2answers
253 views

Nets and compactness in topological spaces.

I am reading Kelley’s book on general topology. There are a few statements on nets there (chapter 2), but the characterization of compact sets in the language of nets is not given. How should we prove ...
6
votes
3answers
118 views

A problem on sequentially compact and countably compact

Recently I came across a problem as follows "I know that sequentially compact implies countably compact. But can anybody tell me please that the converse is true or false."
7
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3answers
241 views

Fiber bundle is compact if base and fiber are

I want to show that the total space $E$ is compact if the fiber $F$ and the base space $B$ are compact. Let $\pi$ denote the fiber projection. Since every point in $B$ has an open neighborhood $U$ ...
4
votes
2answers
895 views

Closed Bounded but not compact Subset of a Normed Vector Space

Consider $\ell^\infty $ the vector space of real bounded sequences endowed with the sup norm, that is $||x|| = \sup_n |x_n|$ where $x = (x_n)_{n \in \Bbb N}$. Prove that $B'(0,1) = \{x \in l^\infty ...
3
votes
3answers
363 views

$f$ continuous, $f: X \to Y$, $Y$ compact Hausdorff. Is $f(X)$ compact?

The question "Is $f(X)$ compact?" is something that occured to me when attempting the Munkres question. I think $f(X)$ is compact. Let $ \{V_\alpha \}$ be an arbitrary open cover of $Y$ such that $ ...