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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2answers
84 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
64 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
4answers
116 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
48 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
40 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
39 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
39 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 ...
7
votes
2answers
218 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
38 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
42 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 ...
0
votes
0answers
37 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 ...
2
votes
1answer
73 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
65 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
52 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
40 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
66 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
51 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
44 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
55 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
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0answers
22 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
91 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
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] ...
0
votes
1answer
30 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.
3
votes
2answers
73 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
53 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 ...
3
votes
2answers
70 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
75 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
58 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 ...
0
votes
1answer
26 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
24 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
25 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
37 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
26 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
0answers
19 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
votes
1answer
42 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
43 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
28 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 ...
0
votes
2answers
32 views

an open subspace of compact space

It is know that every compact subspace of Hausdorff space is closed and every closed set is compact. So I have a question as folows: is there any compact non-Hausdorff space $X$ such that every open ...
1
vote
0answers
20 views

Stone-Weierstrass on a sequentially compact space

I am unable to prove the Stone-Weierstrass Theorem on a compact metric space via its sequential compactness. If that were possible, one could probably prove the Theorem on a sequentially compact ...
0
votes
1answer
58 views

open subspace of locally compact

It is kown that A closed subspace of a locally compact space is locally compact If $X$ is locally compact Hausdorff and a dense subspace $Y\subseteq X$ is locally compact iff $Y$ is open. From the ...
0
votes
2answers
34 views

Compact sets and Open sets in a metric space

I have from reading up on things understood that open sets in a metric space is not compact. Though I have no clue why. I would like to know why is it they are not compact? I know that a compact set ...
2
votes
1answer
21 views

Annulus containing a circunference

Let $S^1=\{x\in\mathbb{R}^2\mid\lVert x\rVert=1\}$, where $\lVert x\rVert$ denotes the Euclidean norm. I am asked tho prove that if $U$ is an open set, $S^1\subset U$, then there exists ...
1
vote
2answers
43 views

For normed vectorspaces $V$, $A,B \subset V$ if $A$ is compact and $B$ is closed then $A+B$ is closed

I am looking for a 'direct' way to show the following statement: Problem: Let $V$ be a normed vectorspace, show that if $A$ is compact and $B$ is closed then $A+B:= \lbrace a+b \mid a \in A, b \in ...
0
votes
0answers
20 views

When is a bounded set in a metric space contained in a compact set?

If $A$ is a bounded subset of a metric space $(X,d)$ with nearest point property , then is it true that $A$ is contained in some compact set ? If $A$ is a totally bounded set of a metric space $(X,d)$ ...
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3answers
54 views

Compact normed vector space

Let $V$ be a normed vector space.If $V\neq \{0\}$ is it true that our space cannot be compact?
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3answers
39 views

Recursive use of the Axiom of Choice

In a standard proof that any sequence-compact metric space $(X,d)$ has a (finite) $\varepsilon$-net, the approach is the following: Make a sequence $(x_n)$ such that $$ x_{n+1}\notin\bigcup_{i=1}^n ...
1
vote
1answer
57 views

Theorem 4.20(c) in Baby Rudin: Is every continuous function whose domain is an unbounded subset of $\mathbb{R}$ uniformly continuous?

Here is Theorem 4.20 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Let $E$ be a non-compact set in $\mathbb{R}^1$. Then (a) there exists a continuous function on ...
1
vote
2answers
103 views

Theorem 4.20 in Baby Rudin: How is this map not uniformly continuous?

Let $E$ be a bounded, non-compact subset of $\mathbb{R}$, let $x_0$ be a limit point of $E$ such that $x_0 \not\in E$, and let $f \colon E \to \mathbb{R}$ be defined by $$f(x) \colon= \frac{1}{x-x_0} ...
7
votes
2answers
86 views

Compact set and continuous function [duplicate]

Let $(E,d), (E',d')$ be two metric space, and $f:E\rightarrow E'$ an injective function such that the image of any compact set from $E$ is compact in $E'$. How can I prove that $f$ is continuous? ...
2
votes
1answer
36 views

Proving a topological space is separable

I am trying to prove the following statement: Prove that if (X,d) is a compact metric space, then X must be separable. Where separable means the following: We say a topological space is separable ...