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
39 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 ...
4
votes
4answers
77 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 ...
1
vote
1answer
162 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
181 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
votes
1answer
216 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
2answers
80 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 ...
2
votes
2answers
33 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
57 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
votes
1answer
136 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
57 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
63 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 ...
1
vote
1answer
221 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 ...
-1
votes
3answers
90 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.($|X|>1)$ Please suggest me ways on how should I think about this.Its quite sure that $X$ cant be ...
0
votes
0answers
147 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
votes
1answer
84 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 ...
3
votes
2answers
95 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
votes
2answers
69 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 $\mathbb{R}$ ...
3
votes
4answers
223 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
votes
1answer
64 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 ...
4
votes
0answers
53 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
votes
1answer
45 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 ...
1
vote
1answer
62 views

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

$(M,d)$ is a compact metric space and $f:M \to M$ is an bijective function such that $d(f(x),f(y)) \le d(x,y) , \forall x,y \in M$ , then is $f$ an isometry i.e. $d(f(x),f(y)) = d(x,y) , \forall x,y ...
8
votes
2answers
315 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
53 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
50 views

Sequential Compactness: Show that there exists a number $\alpha$ and a sequence of positive integers $a_1, a_2, a_3,…$

Here's the problem: Consider the function $f(x)=\text{cos}(\sqrt{x}e^x)$. Show that there exists a number $\alpha$ and a sequence of positive integers $a_1, a_2, a_3,...$ such that $$ \lvert ...
2
votes
1answer
132 views

Is there an errata for Ahlfors Complex Analysis?

I believe a question is incorrectly worded, but I could be wrong as well. I tried searching for an errata for Ahlfors Complex Analysis but was unable to find one. On page 63, question 2, it ask: ...
1
vote
1answer
216 views

Weak convergence + compactness = strong convergence? [duplicate]

Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$ and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for ...
1
vote
0answers
59 views

Compactness & Continuity - Looking for feedbacks on a specific setting

I am trying to get the implications of the following general setting concerning compact spaces and continuous maps. Any feedback would be greatly appreciated, because I have some difficulties in ...
0
votes
1answer
71 views

Compact set is nowhere dense in $\mathbb{N}^{\mathbb{N}}$

Show that any compact set is nowhere dense in $\mathbb{N}^{\mathbb{N}}$, the set of all infinite sequences. A set $A$ is nowhere dense if the interior of its closure is empty, i.e. ...
0
votes
2answers
113 views

Stabilizer, Cosets, homeomorphism and Compact groups : proving things in The Structure of Compact Groups by Hofmann and Morris

I'm currently struggling trying to prove a few things in the book The Structure of Compact Groups by Hofmann and Morris. The first one would be Proposition 1.10.i (or E1.4) : If the topological ...
2
votes
1answer
56 views

Relative compactness and sequences such that $|x_p-x_q|\geq c$

Let $X$ be a Banach space and $B$ is a bounded subset of $X$. If there exist a constant $c>0$ and a sequence $(x_n)_n\in B$ such that $$|x_p-x_q|\geq c,$$ for all $p,q$ with $p\neq q$, then $B$ is ...
5
votes
1answer
57 views

Is the countable product of co-countable topology Lindelöf?

For $i\in\mathbb{N}$, let $(X_i,T_i)$ be the countable complement topology on $\mathbb{R}$. Let $(X,T)$ be the product topology (not box product). Is $(X,T)$ Lindelöf? That is, does every open cover ...
0
votes
1answer
72 views

Why can't we use closed sets to make covers for compactness?

In particular, what about the real line? If our topology is generated by sets of the form [a,b] or [a,b), why can't we form an open cover of, say, [0,1] with those and be guaranteed a finite subcover? ...
1
vote
0answers
27 views

Minimal conditions for compactness of PDFs

I need to find some set of (minimal) conditions to put on a family of probability density functions with bounded support so that the family becomes compact. (I want to use Sion's theorem, which ...
4
votes
1answer
93 views

Compactness of a set of functions

During lunch break, somebody submitted us this problem today: Let $a$ and $b$ be real numbers and $F:\mathbb R\to\mathbb R$ a continuous function. Let $K=\{u\in C^1([a,b],\mathbb R), ...
3
votes
0answers
26 views

Does having a real valued cauchy sequence on a function in a compact space imply the function is continous on that space?

I had to prove for a homework assignment this function $$ s_n(x) = \sum_{i=0}^n (-1)^i \frac{ x^{2i+1}}{(2i+1)!} $$ is a Cauchy sequence with respect to the sup norm for $$ s_n : [-M,M] ...
0
votes
1answer
51 views

Show (0,1) is not compact [duplicate]

Let $I_n=\left(\frac{1}{n},1\right)$. Show that $(0,1)$ is not compact: show that any finite collection of $\{I_n\}$ will not cover $(0,1)$. Give me a hint.
4
votes
2answers
117 views

“Redundant” finite subcovering of a compact space.

Let $M$ be compact and $\mathcal{U}$ an open covering of M such that each $p \in M$ is contained in at least two members of $\mathcal{U}$. Show that $\mathcal{U}$ reduces to a finite subcovering with ...
0
votes
3answers
67 views

How does one show that $\{ \frac{1}{n} | n \in \mathbb{Z_{>0}}\} $is not compact in the standard topology?

How does one show that $\{ \frac{1}{n} | n \in \mathbb{Z_{>0}}\}$ is not compact in the standard topology of $\mathbb{R}$? I know this is not compact because if we take small enough intervals ...
4
votes
2answers
97 views

What are “finiteness” and “discreteness” when it comes to compact sets?

I recently found this answer by Qiaochu Yuan but I'm not sure what "finiteness" and "discreteness" function are in the context of compactness. I've read What does it mean when a function is finite? ...
0
votes
1answer
83 views

Existential Second Order Logic; Compactness and Löwenheim-Skolem

I'm looking for proofs of Löwenheim-Skolem and Compactness in existential SoL. I've spent a substantial amount of time on google, but can't seem to find anything!
1
vote
2answers
66 views

Can't figure out what's wrong with my proof

I have to decide if it possible to find a set $A\subset \mathbb{R}$ such that: $A$ is not connected nor compact but it is complete. At first, I thought it wasn't possible, and made the following ...
1
vote
1answer
48 views

closure of compact subspace

It is known that If $X$ is a Hausdorff space then every compact subspace of $X$ is closed. Hence closure of compact subspace of $X$ is also compact. My question: is there any a $T_1$ space $X$ such ...
0
votes
1answer
34 views

If $A$ is subspace of topological space $X$ is compact and closure of $A$ is not compact then $X$ is particular point topology

I am looking for a topological space $X$ which if $A\subset X$ is compact but closure of $A$ is not compact. From this Find a topological space X and a compact subset A in X such that closure of A is ...
1
vote
1answer
33 views

Compactness of given subsets of $\mathbb R^n$

Looking for some feedback for solutions to select exercises from a basic Analysis course. All comments welcome! Determine whether or not each subset of $\mathbf{R}^2$ is compact. Briefly justify ...
2
votes
1answer
46 views

Volume of a compact set, not necessarily convex

Looking through my lecture notes, I came across the notion that if a set $X\subset \mathbb{R}^n$ is compact and convex and $vol(X)=2^n$, then by choosing an $0<\epsilon <1$, then $X\subsetneq ...
0
votes
1answer
29 views

Holomorphic functions on a connected and compact domain

Consider the following theorem (see references at the end): If $X$ is a connected and compact complex manifold, then any holomorphic function $f : X \rightarrow \mathbb{C}$ is constant. What about ...
0
votes
1answer
72 views

Using the open cover definition of compactness to show that the set of nilpotent $m \times m$ real matrices is noncompact

Is the set of nilpotent $m \times m$ real matrices compact? I found the proof of this statement, using Heine-Borel theorem on $\mathbb R^n$. Tha'ts quite good. But, is it possible to prove this ...
1
vote
1answer
46 views

Compactness and open sets

I have this small question, if $(E,\tau)$ is a Hausdorff space and $A,B$ two separated compact sets, how to prove the existence of two open disjoint sets $U$ and $V$ such that $B\subset V$ and ...
0
votes
0answers
39 views

an open subspace of locally compact is dense

Let $X$ be locally compact Hausdorff. Then a subspace $A$ of $X$ is dense and locally compact iff $A$ is open. I can prove the necessary condition. But for the sufficient condition, I can not get ...