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.

learn more… | top users | synonyms

0
votes
1answer
24 views

Proving subsets of $l^{\infty}$ are compact

Recently I started reading up on some set theory and metric spaces. I just read about compact subsets and I thought I understood it but in the exercises I'm having difficulty with the following ...
3
votes
1answer
25 views

Is compact $T_1$ topological space hausdorff?

I'm in a middle of a very hard exercise which its goal is to prove that some space is hausdorff, but all I could show is that it is $T_1$. But I can also deduce that it is compact. Is that enough for ...
0
votes
0answers
34 views

Non Satisfiability of disjuction

Problem: If S1,S2 are (possibly infinite) sets of propositional formulas where their union: S1VS2 is not satisfiable, prove that there exists an ψ such that S1|=ψ and S2|=¬ψ. Can we say that if ...
0
votes
1answer
22 views

Is Alexandroff duplicate compact?

Consider the Alexandroff duplicate $X\times_{ad} 2$, the space $X\times 2$ where the points of the form $(x,1)$ are isolated and for each open set $U$ in $X$, $(U\times\{0,1\})\setminus (x,1)$ is ...
0
votes
1answer
291 views

Proving that a Closed Interval Is Compact

My text (Stoll, Introduction to Real Analysis, 2nd Ed) defined that $K$, a subset of $\mathbb R$, is compact if every open cover of $K$ has a finite subcover of $K$. Then, it proceeded to prove that ...
9
votes
2answers
75 views

Is any compact, path-connected subset of $\mathbb{R}^n$ the continuous image of $[0,1]$?

If $f:[0,1] \to \mathbb{R}^n$ is any continuous map, then the image $f([0,1])$ is a compact, path-connected set, which is easy to show using some elementary topology. My question is the converse: ...
2
votes
2answers
60 views

In a Hausdorff space the intersection of a chain of compact connected subspaces is compact and connected

Prove that if $X$ is Hausdorff and $\mathfrak{C}$ is a nonempty chain of compact and connected subsets of $X$, then $\bigcap \mathfrak{C}$ is compact and connected. Here are the definitions which ...
1
vote
1answer
17 views

Bounded set that is not closed nor compact

I am to find a set that is bounded but not closed nor compact. Here are my ideas. Please tell me if any of my logic is flawed. I thank you in advance. Consider the set $A = (0,1)$ where $A \subset ...
2
votes
1answer
338 views

$f:M_1\to M_2$ is continuous iff its graph is compact.

I have a propostion in Introduction to Real Analysis (3rd Ed.) which says: If $M_1$ is compact, a function $f:M_1\to M_2$ is continuous iff its graph is compact. Here $M_1$ and $M_2$ are ...
3
votes
1answer
50 views

Does there exist non-compact metric space $X$ such that , any continuous function from $X$ to any Hausdorff space is a closed map ?

I know that there is a topological space $X$ which is not compact but such that , for any Hausdorff topological space $Y$ , any continuous function $f:X \to Y$ carries closed sets to closed sets . I ...
0
votes
2answers
22 views

Negate a proposition with quantifier?

I'm going over the proof of the theorem stating that "In a metric space, compactness impliess sequential compactness". I'm very likely confusing myself. I have the following proposition: $\forall ...
2
votes
0answers
31 views

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$.

The square $S := [- R, R] \times [-R, R]$ is a compact subset of $\Bbb R^2$. An intuitive approach: Let $S$ be not compact then there is an open cover of which there is no finite sub cover of $S$.Now ...
0
votes
0answers
74 views

Uniform best approximation in Chebyshev/Haar systems and the necessity of compactness of the function domain.

A great deal of Chebyshev/Haar systems are given for intervals $]-\infty,\infty[$, $[0,\infty[$ and other noncompact subsets of $\mathbb{R}$. Nonetheless, the theory of uniform best approximations in ...
1
vote
2answers
512 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
1
vote
0answers
40 views

dual ball of L^1 w*-sequentially compact?

Where could I find a direct proof showing that the dual ball of L^1 is w*-sequentially compact? Since $(L^1)^*=L^\infty$, I mean the unit ball $B_{L^\infty}$ with the topology $\sigma(L^\infty,L^1)$. ...
4
votes
6answers
2k views

Every compact metric space is complete

I need to prove that every compact metric space is complete. I think I need to use the following two facts: A set $K$ is compact if and only if every collection $\mathcal{F}$ of closed subsets with ...
15
votes
2answers
698 views

Understanding Alexandroff compactification

Is the Alexandroff one-point compactification of a locally compact Hausdorff space ($\mathbf{LCHaus}$) a functor to the category of compact Hausdorff spaces ($\mathbf{CHaus}$)? It seems to me that one ...
2
votes
2answers
2k views

Prove that a compact metric space is complete.

I'm reading Intro to Topology by Mendelson. I'm in the section titled "Compact Metric Spaces". The problem is in the title. My attempt at the proof is as follows: Let $\{a_n\}_{n=1}^\infty$ be a ...
1
vote
1answer
14 views

Set of points at which a function coincides with its convexification is compact?

Let $f:[0,1]\rightarrow\ \mathbb{\bar{R}}$, and let $\tilde{f}$ be the convexification of $f.$ (i.e., $\tilde{f}$ is the pointwise supremum of all affine functions that lie everywhere below $f$.) Let ...
0
votes
0answers
52 views

Compactness result

I try to prove this lemma: Let $\mathrm{H}_{\text{comp}}^1(\Bbb R^{\mathrm{N}})$ be the subspace of $\mathrm{H}^1(\Bbb R^{\mathrm{N}})$ of functions with compact support. For each ...
0
votes
2answers
140 views

Compactness Theorem explanation

Compactness Theorem definition: If $T$ is a theory in a first-order language $L$, then $T$ has a model iff every finite subset $S$ of $T$ has a model. A number of questions regarding this ...
3
votes
1answer
230 views

Properties if the one-point compactification of an uncountable discrete space

Let $D ( \tau )$ be an uncountable discrete space, and $\alpha D ( \tau )=D ( \tau )\cup\{\alpha\}$ the one-point compactification of $D ( \tau )$. I want to show that if $U$ is any countably ...
2
votes
0answers
26 views

Proving some property of a set of logical expressions that satisfies some properties

I am stuck at this problem. Let $\Sigma$ be a (finite/ infinite) set of logical expression (I.e. strings of the form $(P\land Q)$ or $\lnot(P\lor \lnot (Q\land R))$ etc.). That satisfies the ...
0
votes
1answer
20 views

Equivalence relation, product and quotient spaces

I have a problem with the following: "Define a relation $\sim$ on $R^2$ by $(u,v) \sim (x,y)$ if and only if both $u-x$ and $v-y$ are integers. Show that for each point $(x,y) \in R^2$ there exists ...
-1
votes
2answers
27 views

Compact Subsets [closed]

I drastically need help with these questions. I have been working on this last problem for hours and do not even know where to start or what I am doing. The questions are: a) Let $K$ be a compact ...
4
votes
1answer
185 views

Proving that a sequence in $L^2(\mathbb R)$ is relatively compact

I have a bounded sequence $\{f_n\}_n$ in $L^2(\mathbb R)$ such that $\mbox{supp } f_n$ is uniformly bounded and $$ \int_{\mathbb R} x^2 |\Theta_n(x) (F f_n)(x)|^2 dx \leq C^2 $$ for all $n$, where ...
1
vote
1answer
42 views

Set of all orthogonal matrices over $\mathbb C$ is compact/not

How to show the fact that the set of all orthogonal matrices over $\mathbb C$ is compact By an orthogonal matrix over $\mathbb C$ I mean a matrix $A$ satisfying $AA^T=I$ and here $A^T=(a_{ji})$ where ...
1
vote
2answers
31 views

why union and Cartesian product of infinitely many compact sets is not compact

I'm aware that the union and Cartesian product of finitely many compact sets is compact, but why we can't generalize it to the union and Cartesian product of infinitely many of them? for example for ...
0
votes
1answer
30 views

Show that $R$ is closed but not sequentially compact.

Show that $R$ is closed but not sequentially compact. Attempt: A subset E of a metric space X is said to be sequentially compact if and only if every sequence $x_n \in E$ has a convergent ...
2
votes
1answer
26 views

Showing a mapping is a Homeomorphism

I am trying to prove that the Stone Cech Compactification map is a homeomorphism. I have most the proof finished, but I am stuck on showing that the inverse function is continuous. Here is what I have ...
1
vote
2answers
363 views

Why is the inverse image of a compact set under a special sort of function compact?

Let $f$ be a continuous closed function from $X$ to $Y$ where $X$ and $Y$ are topological spaces. (Closed means that for any closed set $C$, $f(C)$ is also closed). Suppose that for any $y$ in $Y$, ...
1
vote
0answers
27 views

Characterization of compactness in terms of closed sets

I came across an exercise that asked to characterize compactness in terms of closed sets. This is what I came up with: Claim: $X$ is compact $\Leftrightarrow$ for every set of closed sets ...
4
votes
4answers
69 views

$f :\mathbb N \to \mathbb R$ be the function $f(0)=0 , f(n)=\dfrac 1 n , \forall n >0$;is $\mathbb N$ induced with the metric $|f(x)-f(y)|$ compact?

Let $\mathbb N$ be the set of non-negative integers and $f :\mathbb N \to \mathbb R$ be the function $f(0)=0 , f(n)=\dfrac 1 n , \forall n >0$ , then obviously $f$ is injective , so $d : \mathbb N ...
0
votes
1answer
29 views

closed subset of locally compact

A space $X$ is said locally compact if for any $x\in X$ and for any neighbourhood $U$ of $x$ there is a compact neighbourhood $V$ such that $V\subseteq U$. Does closed subset of locally compact is ...
1
vote
1answer
30 views

Prove that the given subset satisfying the given hypothesis is compact.

Let C be a subset of a compact metric space (X, d). Assume that, for every continuous function h : X → R, the restriction of h to C attains a maximum on C. Prove that C is compact. My attempt: I ...
1
vote
1answer
46 views

Why does countable compactness imply compactness on metric spaces?

By "$E$ is countably compact", I mean that every countable open cover of $E$ has a finite subcover. By "$E$ is compact", I mean that every open cover of $E$ has a finite subcover. Let $M$ be a metric ...
1
vote
3answers
141 views

is union of nested compact spaces still compact?

Stel $D$ a metric space. Let $K_1 \subset K_2 \subset K_3 \subset ...$ a serie of compact sets in $D$. I was wondering if $K = \bigcup_{n=1}^\infty K_n$ is compact too. If we take an open cover of $K$ ...
0
votes
1answer
28 views

Intersection between a compact and a locally compact set

I'm trying to understand Rudin's proof of Pontryagin duality theorem, but I still haven't undersood an argument. (Fourier analysis on groups, p29) Let $G$ be a group and denote $\Gamma =\widehat{G}$ ...
1
vote
2answers
14 views

Compactness argument in SVD existence proof

The classical proof of the existence of the SVD factorization by Trefethen and Bau reports Set $\sigma_1 = \mid\mid A \mid\mid_2$. By a compactness argument, there must be a vector $v_1 \in ...
1
vote
2answers
28 views

Are compact Lie algebras necessarily compact as a set of matrices?

I'm reading through a paper and came across something confusing; my limited experience with Lie theory is a bit of a hindrance: The author starts with a compact set of matrices (in the usual ...
0
votes
1answer
51 views

Can the ball $B(0,r_0)$ be covered with a finite number of balls of radius $<r_0$

Consider an infinite dimensional Banach space $X$. Let $B(0,r_0)$ be the ball with radius $r_0$. We know that the ball $B(0,r_0)$ is not relatively compact, so it is not totally bounded. This implies ...
1
vote
1answer
39 views

$X$ is Frechet Compact iff $X$ is compact.

I have done the proof that $1)\ X$ is Frechet Compact iff $X$ is sequentially compact. $2) \ X$ is sequentially compact iff $X$ is compact. Thus we can conclude that $X$ is Frechet Compact iff ...
5
votes
2answers
54 views

Tychonoff's theorem for $[0,1]^\mathbb{R}$

According to Tychonoff's theorem any uncountable product of compact spaces is compact with respect to product topology. Then $[0,1]^\mathbb{R}$, the space of all functions defined on $\mathbb{R}$ ...
2
votes
1answer
130 views

Closure of a set in a “Topology of finite complement”

Well, I was reading this article by Kelley and when reached the point where he say that $X_a$ is closed in $Y_a$ I had to stop, probably mine is just a stupid misunderstand but can't figure out how to ...
1
vote
1answer
48 views

Proving that the intersection of any compact sets is also compact [closed]

I want to prove the theorem using only the definition of compact set. Is there a way to do this? The compact set is defined on the metric space (definition in Walter Rudin PMA).
15
votes
2answers
1k views

Isometry in compact metric spaces

Why is the following true? If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
6
votes
2answers
174 views

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ , then $f$ is isometry?

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ ; then how to prove that $d(f(x),f(y))=d(x,y) , \forall x,y \in M$ i.e. that $f$ is an ...
0
votes
1answer
176 views

A conjecture on uniform convergence of functions with a compact metric space

So I was having a discussion with a friend about this problem and we have conflicting views. Here it is We let $f_n: E \rightarrow \mathbb{R}$ be continuous functions for $1 \leq n \leq N$ and we ...
0
votes
0answers
9 views

Question about metric spaces(compact, dense) [duplicate]

Prove, that every compact metric space has a countable, and dense sub-set. I don't know how I should prove this, I tried with the definition: A topological space X is called compact if each of its ...
0
votes
0answers
39 views

Gromov compactness theorem

Reference: this book, page 493. For a compact metric space $X$ define $\text{Cov}(X,\epsilon)= \min \{n \, : \, X \text{ is covered by $n$ closed } \epsilon\text{-balls} \}$ and ...