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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Prob. 3 (b), Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How does the $K$-topology on $\mathbb{R}$ differ from the usual topology?

Let $$ K \colon= \left\{\ \frac{1}{n} \ \colon \ n \in \mathbb{N} \ \right\},$$ and let the $K$-topology on $\mathbb{R}$ be the one having as basis all open intervals $(a,b)$ and all sets of the form ...
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1answer
24 views

Compactness of a convex collection

Given $\epsilon\in(0,1)$, suppose we have collection $\mathscr{C}(\epsilon)$ of multilinear polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ is in range $[-\epsilon,\epsilon]$ on $S_0$ while ...
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4answers
21 views

Show that if $X$ is sequentially compact, then $X$ is complete and totally bounded

Given a metric space $X$ which is sequentially compact (i.e every sequence has a converging subsequence), show that $X$ is complete and totally bounded. I've already shown that $X$ is complete, since ...
2
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1answer
28 views

Show that if $(X,d)$ is compact then, every open covering of $X$ has a Lebesgue number.

Let $(U_i)_{i \in I}$ be an open cover of a metric space $(X,d)$, a number $\epsilon >0$ is called a Lebesgue number of $(U_i)_{i \in I}$ if for all $x \in X$ exist $j \in I$ such that ...
2
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0answers
33 views

If a set is Hausdorff relative to one topology, can it be compact relative to a strictly finer topology?

Let $\tau_1$ and $\tau_2$ be two topologies on a non-empty set $X$ such that $(X, \tau_1)$ is Hausdorff and $\tau_1 \subsetneq \tau_2$. Can $(X, \tau_2)$ be compact? My effort: Suppose that ...
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0answers
19 views

Metric spaces and compactness [on hold]

Let $X$ be a metric space. If for all compact $K$, the set $K\cap F $ is closed, then $F$ is closed.
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0answers
30 views

How is this convex set compact as well?

Given $\epsilon\in(0,1)$, supposing we have a collection $\mathscr{C}(\epsilon)$ of polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ takes on value $0$ on $S_0$ while being in range ...
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2answers
31 views

Three questions from σ-compact spaces and topological groups

every locally compact subgroup of a Hausdorff group is closed. A Hausdorff and $σ-$compact space X is a Baire space if and only if the set of points at which is $X$ is locally compact is dense in ...
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0answers
21 views

Prob. 1, Sec. 27 in Munkres' TOPOLOGY, 2nd ed: How to show that the compactness of every closed interval implies the least upper bound property?

Let $X$ be an ordered set in which every closed interval is compact. Then $X$ has the least upper bound property. How to prove this? My effort: Let $A$ be a non-empty subset of $X$ such that $A$ is ...
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2answers
52 views

Bounded complete metric space is compact?

This question may seem trivial, but in topology we were taught that in a complete metric space, a subset of that space was compact if and only if it is closed and bounded. Moreover, we are told that ...
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1answer
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Proof cube & trapezium a Compact Space & $E^n$ & $I^n, I^{\infty}$ are connected space ??? [on hold]

I need a serious help here please! Question 1: Prove that $E^n$ & $I^n$, $I^{\infty}$ are connected spaces. After a lot of search I found some two theorems in James Dugundji book. But I still ...
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3answers
62 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
15 views

Additive function in $\mathbb{R}^n$ is continuous, and related subspaces compact

I want to show that the function: $A: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n, (x, y) \mapsto x + y$ is continuous. Also, why is it that if $K, L$ are compact subspaces of $\mathbb{R}^n$, ...
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1answer
26 views

Connection between one-point compactification of $\mathbb{R}$ and $S^1$

Definition I got: A one-point compactification of $X$ is $\hat{X}=(X\cup\{\infty\},\tau).$ The new topology $\tau$ is generated by open subsets of $X$, and $U\cup\{\infty\}$ where $U=K^C$ where $K$ ...
2
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1answer
41 views

Is there an example of a non compact operator whose square is compact?

Is there an example of a non compact linear operator T from a Banach space X to itself such that T^2 is compact? Of course the converse is true, as T ^2 is compact if T is. Here T^2 means T composite ...
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1answer
37 views

How to prove a function is continuous on a compact set?

I´m struggleing with this problem: I know by theorems that inf(d(a,b)) exists if the real value function d is continuous on the set AxB. But how can I prove that d is continuous?
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1answer
59 views

$A$ and $B$ compact in a Hausdorff space implies $A\cap B$ is compact [closed]

Prove that if $A$ and $B$ are compact subset of a Hausdorff space $X$, then $A$$\cap$$B$ is compact.
2
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1answer
24 views

What does compactness of $\mathbb R$ under one of these topologies imply about compactness under the other?

Let $\tau ,\tau_1$ be two topologies on the set $\mathbb R$ .Suppose $\tau \subset \tau_1$ .What does compactness of $\mathbb R$ under one of these topologies imply about compactness under the other? ...
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1answer
52 views

When a sigma-finite space is a sigma-compact space?

$X$ is a topological space, $m$ is a $\sigma-$finite measure on $B(X)$, and what condition can make $X$ be a $\sigma-$compact space? This question is from topological groups (for me). Locally compact ...
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0answers
32 views

A confusion of a real analysis online lecture: Relative compactness

https://www.youtube.com/watch?v=kkKfRaI-cqs At 13.00 what does the professor mean to let those subcovers be "restricted" to Y? Is that a process like A is contained by B implies that A intersect C in ...
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0answers
14 views

core-compact but not locally copact

A space $X$ is called core-compact if the set of all open set in $X, \mathcal{O}(X)$, is a continuous poset. It is known that every locally compact is core-compact. Here, a space $X$ is locally ...
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2answers
32 views

continuous image of a locally compact space is locally compact

Is continuous image of a locally compact space is locally compact? Let $X$ be locally compact(l.c.).Let $f:X\to Y$ is continuous and surjective. A space $X$ is locally compact if for each $x\in X$ ...
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2answers
43 views

compactness of Hilbert cube

I want to show that the Hilbert cube which is: $H=\{(x_1,x_2,...) \in [0,1]^{\infty} : for \ each \ n \in \mathbb{N}, |x_n|\leq \dfrac{1}{2^n}\}$ is compact with respect to the metric: ...
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0answers
25 views

Prob. 7, Sec. 26 in Munkres' TOPOLOGY, 2nd ed: How is the projection onto the first factor closed in 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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1answer
30 views

Arbitrary intersection of closed compact sets is compact (Topology)

Arbitrary intersection of closed compact sets is compact We've been trying to find a counter example to this, however we failed. So we would be happy if someone can tell us if this proposition is ...
9
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1answer
87 views

Epimorphisms of locally compact spaces

Let $LCH$ be the category of locally compact Hausdorff spaces with proper continuous maps. Question. What are the epimorphisms in $LCH$? I suspect them to be surjective, but I haven't been able to ...
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2answers
32 views

Product of Compactly Generated and Locally Compact is Compactly Generated

I'm trying to prove that if $X$ is compactly generated and $Y$ is T2 (Hausdorff) and locally compact then $X\times Y$ is compactly generated. First it is clear that since both $X$ and $Y$ are T2 then ...
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1answer
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Construction of a given neighbourhood in a locally compact group

Let $G$ be a locally compact group. Why is it possible to select a compact neighbourhood $U$ of $e \in G$ such that $U=U^{-1}$ and $gU^2 \subset V$? This is a construction quickly stated by Helgason, ...
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1answer
30 views

Let $K_1 \supset K_2 \supset… $ be a sequence of connected compact subsets of $ \Bbb R^2 $. Is $ K = \cap_{i=1}^\infty K_i $ is connected? [duplicate]

I have managed to write down two proofs showing the connectedness of $K$. But still shaky about both of them. Here are the proofs: 1)Suppose $K$ is disconnected. Then we write it's separation as $ K ...
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2answers
535 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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0answers
15 views

About submanifolds, diffeomorphisms, compact subsets and jordan measurable subsets.

I am given the following problem set which left me pretty puzzled. Let $M \subset \mathbb{R}^n$ be a $C^1$ submanifold and $$\psi : \Omega \rightarrow U \cap M$$ a chart of $M$. $\Omega \subset ...
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4answers
37 views

Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact

Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact under the following metrics: $1. d(f,g)=\sup_{x\in [0,1]}|f(x)-g(x)|$ $2.d(f,g)=\int _0^1 |f(x)-g(x)| dx$ My try: In order to ...
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1answer
32 views

Show that $\exists a\in A; b\in B$ such that $d(a,b)=d(A,B)$

Let $A,B$ be two compact subsets of $X$ where $(X,d)$ is a metric space. 1.Show that $\exists a\in A; b\in B$ such that $d(a,b)=d(A,B)$ where $d(A,B)=\sup\{d(a,b):a \in A;b\in B\}$ 2.Show that ...
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3answers
45 views

Topology: Continuous bijective function, domain = covering compact

Final in real analysis coming up. I could really use some help. If a function f from one set M to another set N is a continuous bijection and M is covering compact, can anything in general be said ...
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0answers
23 views

compact convergence for a series in complex space

I need some help with this. I have to show that the follwing series converges compat. $$\sum_{n=1}^\infty f_n :D:= \{z \in \mathbb{C} | Re(z) > 0 \} \to \mathbb{C}, f_n (z):=\frac{1}{z+n^2} $$ I ...
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1answer
27 views

function on k-space

A topological space $X$ is called k-space if the following condition holds: $A\subseteq X$ is open in $X\iff A\cap K$ is open in $K$ for any compact subest $K$ of $X$. A space $kX$ is a topological ...
6
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1answer
163 views

Lindelöf if and only if every collection with the countable intersection property has non-empty intersection of closures

I am trying to study for my topology exam, and my professor recommended this question from the text (Munkres's Topology (2nd edition), Section 37 question 2): A collection $\mathcal{A}$ of subsets of ...
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1answer
22 views

Image of a bounded sequence by a convex continuous function in a Banach space

Let $(X, \Vert \cdot \Vert)$ be a Banach space, and $f : X \longrightarrow \mathbb{R}$ a convex function, continuous for the norm topology. Suppose that $x_n$ is a sequence which weakly converges to ...
2
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1answer
32 views

compactly generated spaces

A topological space $X$ is called compactly generated if following condition holds: $A\subseteq X$ is open in $X$ iff for every compact $K\subseteq X$, $A\cap K$ is open in $K$. My lecturer said ...
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6answers
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What should be the intuition when working with compactness?

I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow. The point is that I think I'm not really getting the idea on compactness. I mean, in ...
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11answers
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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 $[a,b]$ ...
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1answer
34 views

Maximum and minimum of a function from $\mathbb{R}^n$ to $\mathbb{R}$

Let $A \in \mathbb{R}^{n \times n}$ be a real $n \times n$-matrix. Consider the function $$q: \mathbb{R}^n \to \mathbb{R}, x \mapsto x^t A x$$ where $x^t$ is the transposed vector $x$. I now want ...
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1answer
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unitalization of the $c^*$-algebra of complex polynoms without constant term / compactness of the spectrum of elements in non-unital $c^*$-algebras

Let A be $C^*$-algebra with unit $e$ and $a\in A$ normal. We define $$alg(a,a^*)=\overline{ \{ \sum\limits_{k,l=0}^n\lambda_{k,l}a^k\overline{a}^l; \lambda_{k,l}\in\mathbb{C}, n\in\mathbb{N}\} \\}$$ ...
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3answers
37 views

Prob 10 Sec 26 in Munkres' TOPOLOGY, 2nd ed: How to give examples of this result failing?

Let $X$ be a compact topological space. Let $f_n \colon X \to \mathbb{R}$ be a sequence of continuous functions such that $f_n(x) \leq f_{n+1}(x)$ for all $x \in X$ and for all $n \in \mathbb{N}$. Let ...
2
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1answer
58 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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4answers
53 views

Are there compact manifolds without boundary?

Based on this question I'd like to know: Are there compact (sub)manifolds without boundary in $\mathbb{R}^n$ ? Because, as that question shows, the topology of the manifolds has to be the trace ...
3
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2answers
131 views

Complementary compactness

Let $X$ be a topological space having the property that whenever a subset $A$ of $X$ is compact, then $X\setminus A$ is compact too. Is every subset of $X$ compact?
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3answers
50 views

Compact Sets in $\mathbb{R^{n^2}}$ [duplicate]

I have a question of multivariable analysis and I don't know how to resolve this. The $n \times n$ orthogonal matrices form a compact subset of $\mathbb{R^{n^2}}$? I will be very grateful for ...
4
votes
1answer
41 views

Coincidence points on compact Hausdorff spaces.

I am really stuck on this exercise in my course notes. Let $X$ and $Y$ be compact Hausdorff spaces and $f, g : X \to Y$ be continuous functions. Show that: There is an $x \in X$ with $f(x) = ...
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1answer
21 views

Two questions about increasing unions of compact subsets of a locally compact Hausdorff group.

I have two questions to ask related to my research. Question 1. Let $ G $ be a locally compact Hausdorff group. Is it possible that $ G $ is the union of a chain of compact subsets (ordered by ...