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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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 ...
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0answers
227 views

Subsets of R2 that are convex, closed, and have non-empty interiors?

Can someone give me some guidance with this problem? Thanks. Suppose that $A, B \subset \mathbb{R}$ are convex, closed, and have non-empty interiors. Prove that $A, B$ are the closure of their ...
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1answer
30 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 ...
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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}} , ...
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3answers
92 views

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact.

Let $A,B$ be compact subsets of $X$. Prove that $A \cap B$ is compact. Attempt: Suppose by contrapositive, that $A \cup B$ is compact. Then let $V$ be an open cover of $A \cup B$. Then let $A$ be ...
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3answers
130 views

Is the cone locally compact

Let $X$ denote the cone on the real line $\mathbb{R}$. Decide whether $X$ is locally compact. [The cone on a space $Y$ is the quotient of $Y \times I$ obtained by identifying $Y \times \{0\}$ to a ...
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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 ...
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2answers
558 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 ...
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1answer
54 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 ...
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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 ...
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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 ...
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1answer
31 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 ...
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2answers
74 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
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2answers
55 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 ...
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0answers
35 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 ...
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2answers
136 views

Question about Čech-Stone compactification

Let $\beta X$ be the Čech-Stone compactification of $X$ and $p\in \beta X\setminus X$. Is it true that $\{p\}$ can not be a $G_\delta$ set ?
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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 $A$ ?
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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 ...
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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 ...
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1answer
67 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 $X$ ...
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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 ...
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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 ...
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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 ...
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5answers
406 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 ...
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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 ...
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3answers
413 views

Nets and compactness in topological spaces.

I am reading Kelley’s book on general topology. There are a few statements on nets there (chapter 2), but the characterization of compact sets in the language of nets is not given. How should we prove ...
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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 ...
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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, ...
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1answer
71 views

Prob. 5, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: Every compact Hausdorff space is a Baire space

This is problem 5 in section 27 of Munkres' TOPOLOGY, 2nd ed Let $X$ be a compact Hausdorff space; let $\{A_n\}_{n\in \mathbb{N}}$ be a countable collection of closed sets of $X$. If each set $A_n$ ...
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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
75 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 ...
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3answers
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Cover of (0,1) with no finite subcover & Open sets of compact function spaces

I just got back from my exam and these questions' solutions eluded me, it would be great to use the rest of my evening figuring these out... Q1: Find an open covering of the set $(0,1) \subset ...
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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 ...
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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 ...
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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 ...
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2answers
412 views

Continuous one-to-one mapping of a compact space

Prove that every continuous one-to-one mapping of a compact space is topological. Does this problem statement refer to a mapping of a compact space to itself? If so, suppose the mapping is ...
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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. ...
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3answers
101 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 ...
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1answer
83 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 ...
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0answers
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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 ...
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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 ...
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2answers
39 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 ...
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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.
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
516 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
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1answer
558 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 ...