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
0answers
16 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 ...
2
votes
0answers
25 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
41 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. ...
6
votes
3answers
93 views

Prob 12, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to show that the domain of a perfect map is compact if its range is compact?

Let $X$ and $Y$ be topological spaces such that $Y$ is compact, and let $f \colon X \to Y$ be a closed, surjective, and continuous map such that, for each $y \in Y$, the inverse image $f^{-1} ( \ \{y ...
3
votes
1answer
79 views

Prob 9, Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to prove the generalised tube lemma?

The tube lemma is as follows: Let $X$ and $Y$ be topological spaces. Let $Y$ be compact. Let $x \in X$. If $N$ is an open set in $X \times Y$ such that $x \times Y \subset N$, then there is an open ...
1
vote
0answers
36 views

Prob. 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How is the projection onto the first factor closed if the second factor is compact?

Let $X$ and $Y$ be topological spaces such that $Y$ is compact. Then how to show that the projection map $\pi_1 \colon X \times Y \to X$ is a closed map? My effort: Let $C$ be a non-empty closed ...
1
vote
3answers
43 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 ...
2
votes
2answers
38 views

Condition that a local homeomorphism be a covering map.

Let be $f:Y\to X$ a local homeomorphism, with $Y$ a compact space and $X$ a Hausdorff connected space. How can I show that, for each $x\in X$, $f^{-1}(x)\subset Y$ is finite? So, is clear that $f$ is ...
0
votes
1answer
28 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
27 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
74 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
57 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
56 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 ...
0
votes
1answer
324 views

Proving that a Closed Interval Is Compact

My text (Stoll, Introduction to Real Analysis, 2nd Ed) defined that $K$, a subset of $\mathbb R$, is compact if every open cover of $K$ has a finite subcover of $K$. Then, it proceeded to prove that ...
14
votes
4answers
6k views

Projection map being a closed map

Let $\pi: X \times Y \to X$ be a projection map where $Y$ is compact. Prove that $\pi$ is a closed map. First I would like to see a proof of this claim. I want to know that here why compactness is ...
1
vote
1answer
506 views

Projection mapping closed in compact space [duplicate]

Consider a topological space $(X, \mathcal{T})$. Suppose $X$ is compact and $(Y, \mathcal{T}_Y)$ is Hausdorff. Let $\Phi: X \times Y \rightarrow Y$ be the projection map. We show that $\Phi$ is a ...
3
votes
1answer
553 views

Why is the set $ \mathbb{Z}_{+} \times \{a, b \}$ limit point compact?

I'm having trouble with an example from Munkres dealing with limit point compactness. The example is as follows: Let $Y$ consist of two points; give $Y$ the topology consisting of $Y$ and the empty ...
2
votes
1answer
24 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 ...
4
votes
1answer
115 views

Finding an open set for a topological group

Let $G$ be a locally compact topological group, $K$ a compact subgroup and $\Gamma$ a discrete subgroup. I try to find a neighbourhood $U$ of the identity such that $\Gamma \cap UK = \Gamma \cap K$. ...
13
votes
2answers
394 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 ...
0
votes
1answer
14 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
73 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 ...
1
vote
2answers
551 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
20
votes
11answers
5k views

How to prove $[a,b]$ is compact?

Let $[a,b]\subseteq \mathbb R$. As we know, it is compact. This is a very important result. However, the proof for the result may be not familar to us. Here I want to collect the ways to prove $[a,b]$ ...
1
vote
0answers
21 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
vote
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?
3
votes
2answers
36 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
vote
1answer
49 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 ...
1
vote
1answer
25 views

Why is this image sequentially compact? [duplicate]

Assuming $X$ and $Y$ are normed spaces, $K\subset X$ and $f:K\rightarrow Y$. Why is the image $f(K):=\{f(x)\in Y: x\in K\}$ sequentially compact, if $K$ is sequentially compact and $f:K\rightarrow Y$ ...
12
votes
1answer
231 views

Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected.Then show that $\partial D$ is connected

I am trying to prove the following famous result in Point Set Topology. Let $D$ be a bounded domain (open connected) in $ \mathbb C$ and assume that complement of $D$ is connected. Then show that ...
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 ...
0
votes
1answer
19 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 ...
1
vote
1answer
39 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
99 views

Alternate proof for Arzela-Ascoli

Im trying to finish a beautiful excercise, which consist of giving an alternate proof for the following corollary of Arzela-Ascoli´s Theorem. Given $X,Y$ metric spaces, $X$ compact, $Y$ complete, and ...
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
41 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
23 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 ...
0
votes
1answer
22 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
vote
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 ...
1
vote
1answer
26 views

For compact $A$, $\inf\{\varrho(y,x) : y \in A\}=\varrho(a,x)$

I need help with prooving that if non empty $A$ $\subset(X,\varrho)$ is compact, then: $(\forall x \in X) (\exists a \in A) \inf\{\varrho(y,x) : y \in A\}=\varrho(a,x) $ I found this solution: ...
2
votes
2answers
40 views

Is the intersection of two locally compact subspaces locally compact?

Taking locally compact as such that every point has a local base of compact neighborhoods, is the intersection of two locally compact subspaces locally compact?
0
votes
1answer
22 views

Have you ever seen this result about pointwise/uniform convergence of a net of continuous functions?

I am in need of results transforming pointwise convergence of functions into uniform convergence. Since I wasn't satisfied with Dini's theorems, I had to prove the following result: Let $K$ be a ...
0
votes
1answer
25 views

Can a countably infinite compact topological space have isolated point? Can it admit a minimal subsystem?

Examples I could think of are all sequences with their limit. But is every countably infinite compact space admit atleast one isolated point?
2
votes
1answer
23 views

intersection of two relatively compact spaces

It is known that intersection of two compact spaces is might not compact but intersection of two compact Hausdorff spaces is compact. I curious about intersection of two relatively compact spaces. In ...
0
votes
0answers
28 views

Countable fundamental system of neighbourhoods in a compact Hausdorff space?

Is it true (or false) that every point in a compact Hausdorff-Space has a countable local base, i.e. a countable fundamental system of neighbourhoods? If this is false, which additional property ...
0
votes
0answers
27 views

Arzela ascoli theorem, question?

I have a quick question, in the proof of the Arzela Ascoli theorem one uses the fact that $X$ in $C(X)$(the space of continuous function $X\rightarrow \Bbb C$) is separable. But I don't really see ...
3
votes
1answer
62 views

Finding Function's Extension and Its Unique Existence.

Let $$A= \left\{\frac j{2^n}\in [0,1] \mid n = 1,2,3,\ldots,\;j=0,1,2,\ldots,2^n\right\} $$ and let $$ f:A\rightarrow R $$ satisfy the following condition: There is a sequence $ \epsilon_n \gt 0 $ ...
2
votes
1answer
46 views

Comparing the Samuel and Stone-Čech compactifications of a Hausdorff topological group

Let $G$ be an Hausdorff topological group and let $\beta G$ be the Stone-Čech compactification of $G$. Now, $G$ is also a uniform space with respect to the so-called right uniformity. Let $S(G)$ be ...
12
votes
2answers
126 views

Is $\mathbb{R}^n$ properly homotopy equivalent to $\mathbb{R}^m$ if $n \neq m$?

$\DeclareMathOperator{\id}{id} \newcommand{\R}{\mathbb{R}}$ If $f,g : X \to Y$ are two maps (all maps considered are continuous here), a homotopy between $f$ and $g$ is a map $H : [0,1] \times X \to ...
16
votes
2answers
376 views

Topological spaces in which every proper closed subset is compact

Let $X$ be a topological space. It is a basic result that that if $X$ is compact, then every proper closed subset $Y \subset X$ is compact. Out of curiosity, I would like to explore the converse of ...