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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13
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3answers
369 views

Is a space compact iff it is closed as a subspace of any other space?

I am trying to come up with an alternate definition of a compact topological space that coincides with the usual one. Sorry if my topology is a little rusty. My proposed alternative definition is ...
1
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1answer
32 views

Prob. 2, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: Compactness of $[0,1]$ in the lower limit topology

Let $\mathbb{R}_l$ denote the set of real numbers with the topology having as a basis all the half open intervals $[a,b)$ on the real line. Then is the closed interval $[0,1]$ compact as a subspace ...
0
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0answers
25 views

Prob. 1, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: An infinite subset of $[0,1]^\omega$ without limit points in the uniform topology?

Let $[0,1]^\omega$ denote the set of all sequences of real numbers in the closed unit interval $[0,1]$, and let the uniform metric $d$ on $[0,1]^\omega$ be given by $$d\left( (x_n)_{n\in\mathbb{N}} , ...
1
vote
1answer
56 views

Can anyone provide a proof that a compact set in metric space $(X,d)$ is bounded using..

using anyone of the following definitions(and no other concerning compactness): -$A \subseteq (X, \tau)$ is compact if for every open cover of A there exists a finite cover. -A compact set in a ...
1
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1answer
32 views

Need help with this question concerning compact spaces

Let the set be given like in the following manner: $$\{x_n: n\in\mathbb N\}\subset \mathbb{R^n}$$ $$l^2=\left\{\{x_{n}\}_{n=1}^{\infty}\,\Big|\, \sum_{n=1}^{\infty}|x_n|^2<\infty\right\}.$$ Prove ...
1
vote
2answers
75 views

Does anyone understand this proof: If $A$ is closed and bounded $\implies A$ is sequentially compact.

This is how it goes, I will highlight the parts in yellow which I don;t understand why it is , or the idea behind it. $A$ is bounded so $(\forall x \in A)(\exists M > 0)(\|x\|<M)$ Let ...
4
votes
2answers
56 views

Stone-Čech compactification $\beta\mathbb{N}$ of the integers $\mathbb{N}$ with discrete topology has uncountably many points?

How do I show that the Stone-Čech compactification $\beta\mathbb{N}$ of the integers $\mathbb{N}$ with the discrete topology has uncountably many points? There is a hint that crux is to construct a ...
0
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1answer
68 views

An example for uncountable compact space

Would someone please give an example of a space which is compact but not countably compact space? Is my example right? : suppose there exist a collection of sets ${\{S_i}\}$ for all $i\in \mathbb ...
1
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1answer
54 views

Locally connected Locally compact separable metric space

Let $X$ be a locally connected locally compact separable metric space. Is it possible to find a countable collection $\mathcal{B}$ such that every member of $\mathcal{B}$ is a nonempty peano subspace ...
0
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0answers
36 views

Question regarding open covers, closed intervals and rationals

i wanted to know if open neighbourhoods of arbitrarily small size around the rationals in a closed interval in R can constitute an open cover for the closed interval. I want to use this in a proof in ...
3
votes
2answers
253 views

On the proof of sequentially compact subset of $\mathbb R$ is compact

I don't understand last steps of proving the following theorem: My questions are: 1- Why the statements "cover index $(x_{n_k}) \le$ cover index $(x_0)$ for each index $k \ge K$" or/and "cover ...
2
votes
1answer
58 views

Is every compact set in $\mathbb R^2$ a continuous image of some compact set of $\mathbb R$?

Is it true that for every compact subset $A$ of $\mathbb R^2$ , there exist a compact set $B$ in $\mathbb R$ such that there is a continuous surjection from $B$ to $A$ ?
1
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1answer
44 views

Boundary of a compact set

We have a compact and convex subset $K\subseteq \mathbb R^n$. Also assume that $K$ has more than one point. We want to show that a point $x\in K$ is not on the boundary of $K$ (that is $x\in ...
2
votes
1answer
68 views

Stone-Cech-Compactification

In the lecture, we introduced the Stone-Cech-compactification via ultrafilters. More concretely, we defined $\beta X = \{\mathfrak{U}|\mathfrak{U}$ ultrafilter on $X\}$. This is possible for $X$ ...
0
votes
1answer
33 views

Unit quaternion ball is compact and connected?

Let$$\mathbb{U} := \{x \in \mathbb{H} : |x| = 1\}.$$This is a group under multiplication. What is the easiest way to see that $\mathbb{U}$ is a compact and connected subset of $\mathbb{H}\cong ...
4
votes
2answers
101 views

Showing a set is a compact subset of $\mathbb{R}$

Question: Let $$A=\{ x \in \mathbb{R}: x(x^{3}-3x-1)\leq15 \}.$$ Show that A is a compact subset of $\mathbb{R}$. I am just wondering how to approach this problem. Should I try showing it ...
-1
votes
1answer
37 views

Existence of exhaustion by compact sets

I am wondering when it is known that a set $A$ in topological space $X$ can be exhausted by compact sets, that is there exists increasing sequence of compact sets covering $A$. I guess this should ...
3
votes
2answers
38 views

Prob. 4, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: For $T_1$-spaces countable compactness is equivalent to limit-point-compactness.

Definition: A topological space $X$ is said to be countably compact if every countable open covering of $X$ has a finite subcollection that also covers $X$. Definition: A topological space ...
0
votes
1answer
24 views

Compact set on functions space

Let $(D[0,T], X\times X)$ the set of cadlag functions from $[0,T]$ to $X\times X$. If I have a compact subset $K$ in $(D[0,T], X)$ and another compact subset $H$ in $(D[0,T], X)$, is $K\times H$ a ...
7
votes
5answers
407 views

Necessity of being Hausdorff in the definition of compactness?

According to R Engelking - General Topology: A topological space $X$ is called a compact space if $X$ is a Hausdorff space and every open cover of $X$ has a finite subcover, i.e., if for every ...
1
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1answer
41 views

Example 3, Sec. 28 in Munkres' TOPOLOGY, 2nd ed: How does $S_\Omega$ satisfy the sequence lemma?

Here's the sequence lemma: Let $X$ be a topological space, let $x \in X$, and let $A \subset X$. If there is a sequence of points of $A$ converging to $x$, then $x \in \overline{A}$. Conversely, ...
4
votes
1answer
31 views

Converse of closed graph theorem in general topological space

I was reading this question which shows that for metric spaces, $$M \text{ compact} \iff \big((\text{Graph}(\varphi) \text{ closed} \implies \varphi \text{ continuous}) \,\, \forall \text{ set maps ...
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0answers
33 views

Example of open cover of (0,1) which has no finite subcover [duplicate]

Give an example of an open cover of the segment $(0,1)$ which has no finite subcover. Example: Taking $G_n=(0,1-1/n)$ for $n>1$. It is obvious that $(0,1)\subset \cup_{n=2}^{\infty}G_n$ but ...
3
votes
1answer
31 views

Compact set of real numbers with countably many limit points.

Construct a compact set of real numbers whose limit points form a countable set. My example: Let $E_1=\{1\}\cup \{1+1/n: n\in \mathbb{N}\},$ $E_2=\{1/2\}\cup \{1/2+1/n: n>2\},$ $E_3=\{1/3\}\cup ...
19
votes
1answer
248 views

Let $f:K\to K$ with $\|f(x)-f(y)\|\geq ||x-y||$ for all $x,y$. Show that equality holds and that $f$ is surjective. [duplicate]

$K$ is a compact subset of $\Bbb R^n$ and $f:K\rightarrow K $ satisfies : $$\|f(x)-f(y)\|\geq \|x-y\|$$ Show that $f$ is bijective, and that : $$\|f(x)-f(y)\| = \|x-y\| $$ It's easy to show that ...
0
votes
0answers
76 views

non compact closed range operator

Lately I've been reading Abramovich and Aliprantis' book 'An invitation to operator theory', chapter 2 (page 69) on bounded below operators. I would like to find an example of non-compact (and ...
5
votes
2answers
142 views

$gf$ closed with compact fibers $\implies f$ closed with compact fibers

Call a continuous function $\phi: A \to B$ universally closed if $\phi \times 1_T$ is closed for every topological space $T$. Exercise 3.6.13(d) of Ronnie Brown's Topology and Groupoids asks the ...
1
vote
1answer
45 views

Equivalence of “sequence that admits a cauchy subsequence”

Let $S$ be a subset of a metric space $(X,d)$. I have read (here) the "Sequential characterization of totally bounded subsets" that says the following are equivalent: 1.) $S$ is totally bounded. ...
1
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3answers
51 views

Proof that boundedness of continuous Real Valued functions implies Compactness

I'm looking to prove the following : Let $(X,d)$ be a Metric Space If every continuous real-valued function on $X$ is bounded then $X$ is Compact I saw a proof earlier today If instead $X$ is ...
0
votes
1answer
29 views

Open (closed) sets of a locally compact space.

Let $X$ a locally compact space. How do I show that if $A$ is a open (closed) set in $X$ then $A$ is locally compact? Thank you very much.
0
votes
1answer
28 views

Distance of a point to a subset.

Let $(M,d)$ be a metric space. For a subset $A\subseteq M$ we define the distance of a point $x$ to $A$ as $$\alpha_A(x):=\operatorname{dist}(x,A):=\inf_{y\in A}d(x,y)$$ Prove that: ...
2
votes
3answers
77 views

Which of $(-\infty,\infty]$ and $[-\infty,\infty]$ is homeomorphic to $S^1$?

Is it correct to say that $(-\infty,\infty]$ is homeomorphic to $S^1$? or it is $[-\infty,\infty]$? (considering standard topology). Would you please provide some explanation or better a rigorous ...
3
votes
0answers
60 views

Show that $\varphi : L \to \Bbb{R}$ is continuous.

Let $L,K$ be to compact metric spaces, let $f:K\times L \to \Bbb{R}$ be a continuous function. Define $\varphi : L \to \Bbb{R}$ as $\varphi(y)=\sup_{x\in K} f(x,y)$. Show that $\varphi$ is ...
-1
votes
1answer
59 views

Find the limit of $A={\{(\dfrac{\theta-1}{\theta}}, \theta)|\theta=1,2,3,\dots\}$

Question: Find the limit of $A={\{(\dfrac{\theta-1}{\theta}}, \theta)|\theta=1,2,3,\dots\}$? Here, $\left(\dfrac{\theta-1}{\theta},\theta\right)$ is a point in $\mathbb R^2$ expressed in polar ...
2
votes
1answer
25 views

Integral of Laplace-Beltrami operator over a manifold

Consider an equation $$\Delta u=-he^{u}$$ over a compact 2-manifold $M$, where $u\in C^{\infty}(M)$. In paper "Curvature functions for Compact 2-Manifolds" by Kazdan&Warner it is said that ...
13
votes
2answers
430 views

Why we use the word 'compact' for compact spaces?

Considering the definition of compactness in either Analysis or Topology books, or its equivalent definitions (i.e. [It] is compact $\Longleftrightarrow\dots$), I couldn't understand why ...
1
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0answers
22 views

countable dense set of space of continuous functions on a campact set

Let $X$ be a compact metric space. Let $C_+(X)$ be the set of all non negative continuous functions on $X$. Do there exist a countable dense set of $C_+(X)$? I think the answer is affirmative. For ...
1
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0answers
40 views

closedness of compact sets in some topological spaces

Is there any famous axiom on X other than Hausdorffness or axioms leading to Hausdorffness,such that every compact set in X is closed?
0
votes
1answer
16 views

Infimum of the supremum absolute value of a decreasing sequence of subsets of $\mathbb{C}$ with non-empty intersection

Let $K_{n}$ be a decreasing sequence of bounded subsets of $\mathbb{C}$ such that $\cap_{n}K_{n}=K\neq\emptyset$. Let $\lambda_{n}=\text{sup }_{\lambda\in K_{n}}|\lambda|$ and $\lambda_{0}=\text{sup ...
7
votes
1answer
78 views

Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: Why we need continuity to show the result?

Let $f: X\mapsto Y$ be a closed continuous surjective map such that $f^{-1}(y)$ is compact, for each $y\in Y$. Show that if $Y$ is compact, then $X$ is compact. My question is why do we need $f$ to ...
3
votes
2answers
37 views

What is “approximate compactness”? What is an example of an approximately compact set?

I read this: A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a ...
1
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1answer
51 views

An simple example to show that every countably compact space needn't be compact

I am willing to study compact and connected in topological space and apply in other topological spaces. I am a beginner in this subject. Kindly give some examples. I have went through few books but I ...
0
votes
1answer
24 views

How to define metric in the Space of Holomorphic Functions?

I am looking for a proper way to define distane on the space of Holomorphic functions defined on a domain $D$.Does the Montel's Theorem (Given below from Stein's Book) helps to Characterize Compact ...
3
votes
1answer
50 views

Why is $\Bbb R\setminus\{\frac1n\mid n\in\Bbb N\}$ not locally compact?

I have a question: if I take in $(\mathbb{R},|.|)$ the set $A=\left\{\frac1n, n\in \mathbb{N}\right\}$ and I consider the set $B=\mathbb{R}\setminus A$ I want to prove that $B$ is not locally ...
1
vote
1answer
32 views

Existence of a open set between a compact and an open set

Let $M$ be a compact manifold, $K\subset M$ compact, $U\subset M$ open. Does in this case always exist a open set $V\subset M$ such that $K\subset V\subset\bar{V}\subset U$ ?
2
votes
1answer
44 views

Compacts And The Reciprocal Of The Weierstrass Theorem

While I was studying Functional Analysis, this question arised: Let $K \subseteq \mathbb{R}$ be a subset with the propertie that, for all $f$ continuous ($f \in ...
0
votes
1answer
25 views

example of a particular topological group

Can someone give an example of a topological group $G$ that is not Hausdorff but that contains a fundamental system of neighbourhoods of $1\in G$ consisting of quasi-compact subgroups? Thanks in ...
1
vote
1answer
46 views

Point-wise bounded and equicontinuous sequence of functions has a uniformly convergent subsequence

Problem We have a sequence $(f_n)$ of continuous functions on a compact metric space K. It is also given that $(f_n)$ is point-wise bounded and equicontinuous. Now show that $(f_n)$ has a ...
0
votes
1answer
23 views

Two (maybe nonequivalent) definitions of local compactness

$X$ is locally compact if every point has a neighborhood with a compact closure. $X$ is locally compact if every point lies in the interior of a compact subspace of $X$. Clearly, $(1) \implies ...
1
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1answer
41 views

Is every compact totally ordered space homeomorphic to a subset of $[0,1]$?

Let $(X,\leq)$ be a totally ordered set such that, equipped with the order topology, $X$ is compact. Is then $X$ homeomorphic to a closed subset $A \subseteq [0,1]$? A way to ask this question ...