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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2
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1answer
75 views

In a compact space, every net has a convergent subnet

I'm just learning how to work with nets. I'm attempting the proof that $X$ compact $\implies$ every net in $X$ has a convergent subnet, and I wonder if I'm overcomplicating it. Suppose $\langle x_i ...
3
votes
1answer
23 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...
0
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0answers
74 views

alternative Compactness theorem proof

I'm attempting a problem which requires me to prove the compactness theorem for propositional logic ![enter image description here][1]in a slightly different way to normal. I'm struggling to ...
0
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0answers
47 views

How to prove that a metric space is compact if it is complete and totally bounded?

How to prove that a metric space is compact if it is complete and totally bounded? Wiki wrote that it is a generalisation of Heine–Borel theorem but I can't prove it.
1
vote
1answer
40 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
0
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0answers
70 views

Finite covering space with compact spce.

Prove that if $p: \ Y \rightarrow X$ is finite covering, then if $Y$ is compact so it is X. Can someone check my attempt? :) Let $\mathcal{U}$ be any open cover of $X$. For every $x \in X$ let us ...
0
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3answers
62 views

Compact Space: Locally Continuous $\implies$ Uniformly Continuous

Given metric spaces. Prove that any locally continuous function on a compact space is uniformly continuous!
3
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1answer
95 views

Finer topologies on a compact Hausdorff space

If we have such topological space $(X,\mathcal{T})$ that it is compact and Hausdorff, then we can say that for any other topology $\mathcal{H}$ on $X$ such that $\mathcal{T}\subseteq\mathcal{H}$, the ...
4
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1answer
66 views

Compactness and sequential compactness in metric spaces

I got a question: I'm trying to proof that every metric space is compact if and only if the space is sequentially compact. In all the proves I have found, they used the Bolzano-Weierstrass theorem. Is ...
1
vote
1answer
57 views

Prove that a relatively compact subset of $L^p$ is bounded.

Let $p\in [1,\infty)$, $A\subset L^p(\mathbb R^m)$ relatively compact and $\lambda^m$ be the Lebesgue measure on $\mathbb R^m$. Prove: a) $A$ is bounded. b) $\lim_{y \to 0}\sup_{f \in A} ...
1
vote
1answer
68 views

Cardinality of all compact metric spaces

I`m looking for cardinal number of all compact metric spaces. I know that: Cardinal number of compact set is at most $\mathfrak{c}$ (it is a continous image of Cantor set) Compact metric space is ...
0
votes
1answer
48 views

Compact set in $(\mathbb R,\rho_1)$

$P = \mathbb R, \rho(x,y):|x|+|y|$ if $x \ne y $ or $0$ if $x=y$. Question: is $[-1,1]$ in $(P,\rho)$ compact set? I think yes: $[-1,1]$ is bound set, all sequences in it also bound, and by ...
5
votes
1answer
108 views

Property of compact, convex sets in $\mathbb{R}^3$

How to solve the following: Let $K\subset \mathbb{R}^3$ be a convex, compact set with smooth boundary $C=\partial K$ and let $\vec{u}$ be any vector. Show that there exist points $x\neq y$, ...
2
votes
2answers
67 views

Naive question about the group $SU(n)$?

As usual, let $SU(n)$ represent the set of all the $n\times n$ unitary matrices with determinant $1$. It's easy to show that any matrix $U$ takes the form $U=e^{iA}$ ($A$ is a $n\times n$ traceless ...
2
votes
1answer
50 views

A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$.

Let $f_n \to f$ on compact subsets of the real line. If $u_m \rightharpoonup u$ in $L^2(0,T;H^1) \cap L^p(0,T;L^p)$ and $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in ...
1
vote
1answer
62 views

Using the compactness theorem

I am working through problems which ask you to apply the compactness theorem (from propositional logic) to problems. How would you go about solving this one? Let $\mathbf{L}$ be an arbitrary ...
0
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0answers
25 views

Stone-Čech compactification not by ultrafilters only.

I am familiar with Stone-Čech compactification using ultrafilters. But, I, somehow can't understand the construction by commutative diagram, and certainly can not see the connection between the two ...
0
votes
1answer
37 views

For compact subspaces $C$ and $K$ of $X$ and $Y$, prove that for every open set $U$ of $X \times Y$, there exist open sets $V$ and $W$ with…

Let $C$ be a compact subspace of $X$ and let $K$ be a compact subspace of $Y$ . Let $U$ be an open set in $X \times Y$ containing $C \times K$. Show that there exist open subspaces $V$ of $X$ ...
0
votes
0answers
40 views

Covering dimension of a compact metric space

I would like to see the proof of the following fact (references appreciated). A compact metric space $X$ has covering dimension $\leqslant n$ if and only if there is a continuous surjection $\pi ...
0
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0answers
25 views

compactness, real analysis

I need help with this excercise, I got a), and c must follow from the extreme value theorem, and the last answer in c is yes? My main problem is with b). It seems that I should start with an open ...
1
vote
1answer
33 views

Approximation by finite sets

I'm reading the book "Topology and Order" by L.Nachbin. In chapter $3$ he speaks about properties of compact Hausdorff spaces. He writes: [A]lthough these spaces may be infinite, they admit ...
5
votes
2answers
47 views

Is the set of translations of a function compact?

Let $X=BUC(\mathbb{R})$ be the Banach space of real bounded uniformly continuous functions on $\mathbb{R}$ equipped with the supremum norm. Let $f\in X$, then the subset $$\{f_a:t\mapsto f(t+a), \ \ ...
3
votes
2answers
38 views

How to “convert” from net to sequence in a first countable space

In a first countable space, what's a good way of going from nets to sequences? Let me explain more clearly what I mean. Suppose $f:X\to Y$ is a topological map and $X$ is first countable. Then I ...
0
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0answers
86 views

An infinite compact set which allows no boundedness and analyticity

I need an example of an infinite compact set $K$ in $\mathbb {C}$ such that there does not exist any non-constant function which is both bounded and analytic on $\mathbb{C} - K$. First, any hints ...
0
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1answer
33 views

How to show $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ is path connected and compact?

let $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ . Show that A is path connected and compact. my attempt: since $\frac {x^2}{9}+\frac{y^2}{4}=1$ is elips. A is bounded and closed. so is compact. (by heine ...
8
votes
2answers
86 views

Topological distinguishibilty of $\infty$ after one point compactification?

Let $X$ be the one point compactification of some locally compact Hausdorff space. Let $\infty \in X$ represent the added point. Is there always a homomorphism $\phi:X \to X$ with $\phi: \infty ...
2
votes
1answer
69 views

Countable union of relatively compact sets

Let $X$ be a topological space and $\mathcal K(X)$ be $\sigma$-algebra, generated by compacts of $X$. Prove that for any set $B \in \mathcal K(X)$ either $B$ or its complement can be represented as a ...
0
votes
0answers
21 views

Jordan content under continuous differentiable map

I have the following problem which seems simple but in fact I find no proof for it so I am wondering if I could get some help. Let $A$ be a compact set subset of an open set $U$ in $\mathbb{R^n}$, ...
1
vote
1answer
67 views

Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.

Say that a topological space is CH if it is both compact and Hausdorff. Let $T$ and $T'$ be two topologies on the same set X that are comparable but different, i.e., $T$ is either strictly ...
2
votes
1answer
45 views

I need a feebly compact topological space that is not pseudocompact

A Tychonoff topological space X is called pseudocompact if every continuous real-valued function with domain X is bounded. A space is called feebly compact if every locally finite family of open sets ...
0
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1answer
36 views

question about Stone Čech compactification

$x$ is normal space and we recognize him by his picture in $βX$. show that every $c_1 c_2$, close and disjoint sets in $x$ also the closure of $c_1$ and $c_2$ (in the closure of $x$) is disjoint. i ...
3
votes
1answer
50 views

Exapmles for Stone–Čech compactification

I'm finding it a bit though to "feel" this topic of Stone–Čech compactification. For example, I want to show that $[0,1]$ is not a Stone–Čech compactification of $(0,1]$ and on the other hand ...
1
vote
1answer
107 views

Distance between any two points in a compact metric space

I am given the following problem: Show that if a metric space (X,d) is compact (meaning X is compact with respect to the metric d), then there exist points a,b ∈ X such that d(a,b) = ...
0
votes
2answers
35 views

X remains Subspace under Compactification

We just had the definition of compactification in our lecture, which says that Y must be compact and X must be a dense and open subspace of X. However in his Notes he gave the Definition so that X ...
1
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0answers
28 views

Convex sets and convex polytopes?

Consider the set $\mathcal{X} \subset \mathbb{R}^d$ convex and compact. Which is the difference between the collection of compact convex subsets of $\mathcal{X}$ and the collection of convex polytopes ...
0
votes
1answer
38 views

Relations between closed and compact sets

I have the following doubts: consider the set $\mathcal{Y}=[0,1]$ which is closed, convex, compact. Let $\mathcal{F}$ be the collection of closed subsets of $\mathcal{Y}$ and $\mathcal{K}$ be the ...
0
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0answers
36 views

How to prove that a sub-space of the functions $f: X \to Y$ is equicontinuous?

Let $X$ and $Y$ be two metric and compact spaces, and $C(X,Y)$ - the metric space of the continuous functions $f:X\rightarrow Y$. Denote by $Y^X$ the space of all functions (not just continuous) ...
0
votes
1answer
52 views

Do continuous functions preserve limit point compactness when the spaces are Hausdorff?

This is based on the problem in Munkres. Let $X$ be a limit point compact Hausdorff space and $Y$ a Hausdorff space. Let $f: X \to Y$ be continuous. Is $f(X)$ limit point compact in $Y$?
0
votes
1answer
52 views

Relatively compact sets in $\bar{\mathbb{R}}^{d}_{0}$

Reading an article I came across the following line, which botheres me since quite a while. Let $E:=\bar{\mathbb{R}}^{d}_{0}:=([-\infty,0)\cup(0,+\infty])^d$ be the closure of $\mathbb{R}^d$ without ...
1
vote
1answer
111 views

why compact support implies a function vanished at boundaries?

"A function has compact support if its support is a compact set." While support of a function $u:G\rightarrow\mathbb{R}$ is defined to be $supp(u)=\overline{\{x:G|u(x)\neq0\}}$ But lately, Another ...
1
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0answers
37 views

Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
3
votes
2answers
92 views

Direct proof of compactness of $\mathbb{Z}_p$

Let $\mathbb{Z}_{p}$ be completion of $\mathbb{Z}$ with respect to $p-$norms. Actually I know that $\mathbb{Z}_{p}$ is bijective to Cantor set, which is compact, therefore by homeomorphism, it is also ...
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0answers
41 views

Exercise in Section 2.4 of Singer & Thorpe

I'm trying to solve the exercise in Section 2.4 of Singer & Thorpe, which is to prove that if $S$ is a compact Hausdorff topological space and $(U_n)_{n \in \Bbb N}$ be a family of dense open ...
1
vote
3answers
53 views

Compactness, topology

In a general topological space $(X,\tau)$ I have the following situation: $$F\subset M\subset N$$. If I prove that $F$ is compact in $N$ (w.r.t the induced topology), is it true that $F$ is compact ...
-2
votes
1answer
62 views

Closed and Compactness on $\mathbb Q$ (Multiple Choice)

Please help me regarding the following question. Consider $\mathbb Q$ with usual metric (i.e $d(p,q)=|p-q|$).Then which of the following are true? $\{q\in\mathbb Q|2<q^2<3\}$ is closed ...
0
votes
1answer
69 views

Is Alexandroff Duplicate A(X) of X paracompact?

Prove or disprove: If $X$ is a paracompact space, then Alexandroff Duplicate $A(X)$ of $X$ is paracompact. Thanks for any help. ...
3
votes
0answers
138 views

Is every compact space compactly generated?

I am using the definition of compactly generated space from The Category of CGWH Spaces, which is In $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in ...
4
votes
2answers
165 views

Map preserving intervals but discontinuous

Let $f:\mathbb{R}\to\mathbb{R}$ be a map sending closed intervals to closed intervals. Prove that $f$ is continuous or find a counter example. WLOG we just have to prove continuity at $0$ and we can ...
2
votes
0answers
93 views

Proof of uniform continuity on compact sets

Show that a function $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous on a compact set $K$ is uniformly continuous on $K$. Is the proof below correct? Proof: Let $\epsilon > 0$ and let ...
4
votes
2answers
53 views

$\left\{x\in H: 2\leq \|x\|\leq 5\right\}$ is compact?

In a Hilbert space $H$ of dimention infinite, $A=\left\{x\in H:2\leq \|x\|\leq 5\right\}$ is compact? (totally bounded and complete) Thanks in advance.