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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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
47 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 ...
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9 views

Smallest integer $N(\epsilon)$ such that $K\subset \bigcup_{n=1}^{N(\epsilon)}B(x_i,\epsilon)$

In a metric space, a set $K$ is said to be totally bounded if for each $\epsilon>0$ there exist a finite number of balls $B_1,B_2\dots B_{N(\epsilon)}$ with radius $\epsilon$ which covers $K$. ...
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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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14 views

separation in perfectly normal spaces [on hold]

Suppose that we have two distinct points $x$ and $y$ in a perfectly normal compact space. Can we find a norm one continuous function such that $f$ equals to $1$ in some neighborhood of $x$, $f$ equals ...
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15 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
45 views

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

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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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 ...
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1answer
38 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 ...
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1answer
171 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
45 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
35 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 ...
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1answer
45 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
79 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 ...
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2answers
52 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
86 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 ...
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1answer
44 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
34 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 ...
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1answer
34 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 ...
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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 ...
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0answers
18 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 $\Gamma =\widehat{G}$ ...
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56 views

erwin kreyszig introductory functional analysis with applications page 82 exercise 4 [closed]

We know that the space $S$ consists of the set of all (bounded or unbounded ) sequences of complex number and the metric $d$ defined by : $$d(x,y)= \sum_{j=1}^{+\infty} \frac{1}{2^j} ...
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2answers
36 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 ...
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0answers
22 views

Proving version of Stone Weierstrass for locally compact space

Let $X$ be a locally compact Hausdorff (LCH) space. Suppose that $\mathcal{A}$ is a closed algebra of $C_0(X)$ (the continuous real-valued functions on $X$ with compact support). Suppose in addition ...
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1answer
47 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: ...
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1answer
26 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 ...
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0answers
49 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 ...
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1answer
28 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. ...
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38 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 ...
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1answer
48 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
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1answer
40 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 ...
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1answer
51 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? ...
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0answers
15 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 ...
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1answer
86 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), ...
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0answers
25 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] ...
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1answer
23 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.
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42 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 ...
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3answers
52 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 ...
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2answers
59 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? ...
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1answer
63 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!
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2answers
47 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 ...
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1answer
23 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 ...
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1answer
22 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 ...
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1answer
23 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 ...
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1answer
27 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 ...
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1answer
21 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 ...
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0answers
18 views

Show that for an infinite subset M in the space s to be compact

I have to show that for an infinite subset M in the space s to be compact, it is necessary that there are numbers y1,y2,... such that for all x=(Ek(x))is an element of M, we have the absolute value of ...
0
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1answer
33 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 ...
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1answer
42 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 ...