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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Does $\sigma$ -compact imply separable?

Let $D$ be a metric space. If $D$ is $\sigma$-compact, does this imply that $D$ is separable? I thought I had a proof, but I think it is wrong. my proof: Let $K_n$ the compact sets such that $K_n ...
7
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2answers
335 views

Existence of a continuous function which does not achieve a maximum.

Suppose $X$ is a non-compact metric space. Show that there exists a continuous function $f: X \rightarrow \mathbb{R}$ such that $f$ does not achieve a maximum. I proved this assertion as follows: ...
3
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1answer
17 views

compact inverse is compact in canonical homomorphism

Let $G$ be locally compact Hausdorff group. Let $N$ be a closed normal subgroup of $G$. Let $f:G\to G/N$ be the canonical homomorphism. I want to show that for every compact subset $C$ of $G/N$, there ...
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0answers
39 views

Normal space is compact

I know that a compact Hausdorff space implies Normal, but does the converse holds? I.e. If a space is normal, it is compact and Haudorff. (Although $T_4$ imlicitly implies $T_2$)
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0answers
38 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
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1answer
35 views

Spaces in which “$A \cap K$ is closed for all compact $K$” implies “$A$ is closed.”

Let $X$ denote a topological space. For any $A \subseteq X$, consider two possible conditions on $A$. $A$ is closed $A \cap K$ is closed, for all compact $K \subseteq X$. If $X$ is Hausdorff, then ...
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1answer
25 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
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1answer
27 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 ...
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1answer
25 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 ...
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2answers
78 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: ...
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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 ...
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2answers
77 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 ...
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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 ...
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0answers
32 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 ...
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0answers
42 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 $\sigma(L^\infty,L^1)$. ...
3
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1answer
51 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 ...
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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 ...
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0answers
53 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 ...
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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 ...
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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 ...
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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 ...
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2answers
32 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 ...
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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 ...
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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 ...
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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 ...
2
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1answer
29 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 ...
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0answers
29 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 ...
0
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1answer
31 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 ...
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4answers
70 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 ...
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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 ...
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3answers
142 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 $K$ ...
4
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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 ...
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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 ...
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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 ...
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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 ...
2
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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 ...
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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 ...
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1answer
49 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).
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0answers
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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 ...
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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 ...
0
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1answer
177 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 ...
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3answers
58 views

Give an example of a compact metric space $X$ such that $X$ and $X\times X$ are homeomorphic

Give an example of a compact metric space $X$ such that $X$ and $X\times X$ are homeomorphic. Please suggest me ways on how should I think about this.Its quite sure that $X$ cant be finite. I tried ...
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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 ...
2
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1answer
51 views

To prove Heine-Borel theorem for $\mathbb R^n$ with usual Euclidean topology

To prove that any closed and bounded subset of $\mathbb R^n$ is compact , I proceed as : Since $\mathbb R^n$ is complete so any closed subset of it is complete . Then I show that any bounded subset of ...
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2answers
83 views

Find all compact sets in $\mathbb{R}$

In $\mathbb{R}$, considering the topology consisting of the empty set and all sets containing $0$ and $1$, I need to find all compact sets. I understand the definition of a compact set but don't know ...
5
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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 $\mathbb{R}$ ...
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4answers
87 views

There's no continuous injection from the unit circle to $\mathbb R$

I read a proof that goes as follows: Let $U$ be the unit circle, and let $f : U \longrightarrow \mathbb R$ be a continuous mapping. $U$ is compact and connected, so $f(U)$ is a closed, bounded ...
2
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1answer
45 views

A set $A \subset l_1$ is compact

A set $A \subset l_1$ is compact if and only if $A$ is closed and bounded and given any $\epsilon >0$, there exists $n_0$ such that $\sum_{k=n}^{\infty} |x_k| < \epsilon$ for all $n> n_0$ and ...
3
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0answers
35 views

$E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?

Problem The graph $G$ of $f$ is defined as the points $(x, f(x))$ for $x \in E$. Suppose $E \subset \mathbb{R}$ is compact, then $f : E \to \mathbb{R}$ is continuous iff its graph is compact. ...
3
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1answer
35 views

If $X$ is a metric space such that any metric space $Y$ , which is a homeomorphic image of $X$ , is complete , then is $X$ compact? [duplicate]

Let $X$ be a compact metric space , then it is easy to show that every homeomorphic image metric space of $X$ is complete . Is the reverse true ? That is if $X$ is a metric space such that any ...