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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2answers
31 views

Is sequence convergent in subspace of compact metric space?

Problem is as follow. Let X be a compact metric space and A be a closed subset of X. Prove that every sequence in A has a convergent (note: convergent in A) subsequence. It is from my note. My ...
0
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1answer
32 views

Compact and convex discrete set

I am working with discrete sets but I have a doubt: is the set $\{ 0,1\}$ compact and convex? And the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$?
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2answers
135 views

Let $p: E\to B$ be a covering map. If $B$ is compact and $p^{-1}(b)$ is finite, then $E$ is compact. [duplicate]

So I start off and assume that some $\{U_\alpha\}$ is a cover of $E$. I want to reduce this cover to a finite subcover of $E$. Since $p$ is a covering map it is also an open map, therefore ...
0
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1answer
21 views

Redefine a discrete compact set

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,\ldots,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as ...
0
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1answer
23 views

Define a compact and convex set through inequality constraints

I need to find twice continuously differentiable functions $g_i: \mathbb{R}^2 \rightarrow \mathbb{R}$ $i=1,...,I$ such that the set $\{ 0,1\}^2=\{(0,0), (1,0), (0,1), (1,1) \}$ can be written as $\{x ...
2
votes
1answer
46 views

$\beta \omega$ is zero dimensional and extremally disconnected

I have proved that $\beta \omega$ is zero dimensional by constructing it using ultrafilters, but I know that $\beta \omega$ is characterized up to topological equivalence by the extension property, so ...
1
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1answer
54 views

Is the category of these particularly nice spaces cartesian closed?

Is the category of Hausdorff, compactly generated, locally path-connected, semi-locally 1-connected spaces (and continuous maps between them) cartesian closed? If not, in what ways does it fail to be? ...
3
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2answers
70 views

Finite-case symmetry leads to infinite-case asymmetry

Formulas for sines or cosines of sums superficially appear to have a certain symmetry, specifically it looks as if sine and cosine play something like symmetrical roles: $$ \begin{align} & ...
2
votes
2answers
81 views

Stone-Čech compactification using ultrafilters

Let $X = \omega \cup \{ x \}$ ne the Stone-Čech compactification of $\omega$. (I am viewing $X$ as a subspace of the set of ultrafilters over $\omega$). Let, $\mathcal A$, $\mathcal B$ be two disjoint ...
1
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2answers
765 views

Proof that a product of two quasi-compact spaces is quasi-compact without Axiom of Choice

A topological space is called quasi-compact if every open cover of it has a finite subcover. Let $X, Y$ be quasi-compact spaces, $Z = X\times Y$. The usual proof that $Z$ is quasi-compact uses a ...
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3answers
60 views

Developing an intuition for compact and open sets

I'm having trouble picturing what compact sets and open sets actually are. Open and closed intervals make enough sense to me, but for whatever reason, moving to the next level of abstraction is ...
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0answers
54 views

Weak sequential compactness in a reflexive space

Let $\{X, \| \cdot \|\}$ be a normed space, $B$ is the unit ball of $X$. If $\{X, \| \cdot \|\}$ is reflexive, then is $B$ weakly sequentially compact? If it's not true, are there any counterexamples ...
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1answer
205 views

Can we prove that a bounded closed subset of $\mathbb R^n$ is compact without Axiom of Choice?

Can we prove that a bounded closed subset of $\mathbb R^n(n \ge 1)$ is compact without using Axiom of Choice? This is a related question which was closed.
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0answers
42 views

Onto continuous function on a compact metric space is isometry. [duplicate]

Let $K$ be a compact metric space with metric $d$ and suppose $f:K\rightarrow K$ is continuous and surjective (onto), and satisfies $d(f(x),f(y))\leq d(x,y),\,\forall x,y\in K$. How can we prove that ...
3
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0answers
24 views

Surjective function on a compact metric space [duplicate]

Assume $f:K\rightarrow K$, is surjective and $K$ is a compact metric space and we have $d(f(x),f(y))\leq d(x,y)\, \forall x,y\in K$. How can I prove that $d(f(x),f(y))= d(x,y)\, \forall x,y\in K$? ...
4
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1answer
71 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 ...
5
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1answer
81 views

Compact topological space not having Countable Basis?

Does there exist a compact topological space not having countable basis? I have constructed a product space from uncountably many unit intervals $[0,1]$, endowed with the product topology. ...
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0answers
49 views

Why is proof of the [topological] closed graph theorem incorrect?

Specifically, the closed graph theorem I am referring to is: Let $f : X \rightarrow Y$ exist and $Y$ be compact and Hausdorff. Then $f$ is continuous if and only if the graph of $f$ denoted by $G_f = ...
3
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1answer
1k views

A proof about $F_\sigma$, $\sigma$-compact sets, and subsets of the irrationals

I've been looking at a proof that shows the following result. $\mathbb{P}$ is the set of irrational numbers, $\mathbb{Q}$ the rationals, and $\mathbb{R}$ the reals. The following conditions are ...
2
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1answer
48 views

Compactness in Complete Lattices

Let $(L,\leq)$ be a complete lattice. We say an $a\in L$ is compact in $L$ iff for $\forall A\subset L$ such that $a\leq \bigvee A$, there exists a finite subset $A'\subset A$, sucth that ...
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1answer
59 views

A closed bounded set having exactly one accumulation point has the covering property

I need to show that a closed, bounded set having exactly one accumulation point has the covering property. A set has the covering property if any open cover of it has a finite subcover. Since the ...
2
votes
2answers
63 views

Limit point compactness

Let $(X, d)$ be a metric space and $A ⊆ X$. If $A$ is limit point compact, show that $A$ is closed. My thoughts: The definition of limit point compactness is that for each infinite subset of $A$, it ...
3
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1answer
54 views

About the Stone-Čech universal property

There's something I am missing here and I don't know what it is. I understand that the Stone-Čech compactification of $X$ satisfies the property that for every continuous map $f: X \rightarrow K$ ...
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0answers
97 views

understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
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1answer
49 views

Constructing a Set with Connected Interior

Suppose that $K\subset\mathbb C$ is a compact set with non-empty interior and suppose that $a\in\operatorname{int} K$. I want to construct a set $M$ with the following properties: $M\subseteq K$; ...
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0answers
15 views

How to use a base to prove something is sequentially compact.

I know this is not very specific but I'm studying for a topology exam and this is one of the things I need to know how to do. I know that part of the process is showing it converges. I was hoping ...
0
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1answer
15 views

Is (X, U) compact? (X, T)?

Let T and U be topologies on a set X. a.) Suppose (X,T) is compact and T is contained in U. Is (X,U) compact? b.) Suppose (X,U) is compact and T is contained in U. Is (X, T) compact? c.) Suppose ...
2
votes
1answer
75 views

compactness in topology of pointwise convergence

I started reading about the topology of pointwise convergence. So far I do not feel quite comfortable with this theory. Maybe one can help me out in a more concrete example case. Let's consider ...
1
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1answer
28 views

If $\omega$ is compactly supported form then so is $d\omega$?

If $\omega$ is a compactly supported differential form then so is $d\omega$. Is it true?
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0answers
33 views

Show a subset A is compact if and only if the image of the map T(A) is compact

The question is as follows: Let $\{v_1,v_2,\dots,v_n\}$ be a set of linearly independent vectors of an $n$-dimensional normed linear vector space $V$. Define the map $T: R^n \to V$ by $T(a) = ...
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0answers
30 views

Question about finite subcovers

I'm having problems wrapping my head around the part with $\rho_i$.Here goes: $A \subset \mathbb{R}^n$ is compact, $\rho$ is a positive real-valued function defined on $A$. Prove: $\exists$ finitely ...
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3answers
98 views

Closed subset of compact set is compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (a) Prove this using definition of compactness. (b) Prove this using the Heine-Borel theorem. My solution: ...
0
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1answer
67 views

Find an example of a compact space which is not locally compact.

I know that every $T_2$ compact space is locally compact.So I need to find a space $X$ that is compact but not $T_2$ , then prove that the there exist a point $x$ that is not in $A^o$ for $A$ is ...
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1answer
57 views

What does $X^Y$ mean?

I have been reading about the Stone-Čech compactification recently and one way of constructing it is by considering the map $h:X\rightarrow I^{C}:x\mapsto(fx)_{f\in C}$ where $I$ is the closed unit ...
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2answers
40 views

Is the following space compact?

Is the subspace of rational numbers in the usual space of real numbers compact? I'm not exactly sure what this is asking. Is this asking if I can generate a cover using a finite amount of sets from ...
2
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2answers
138 views

Compactness of Topological Spaces

The only one that I have been able to get at is that (c) is not compact since it is not closed/bounded by Heine-Borel Theorem. Any thoughts on how to approach the others? I understand that, in order ...
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2answers
42 views

Finiteness of a compact subset in $\mathbb R^n$

Let $K$ be a compact subset of $\mathbb R^n$ such that for all $x \in K$, $K\setminus\{x\}$ is also compact. Show that $K$ is finite. I'm trying to solve it using sequences, but am having difficulty. ...
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2answers
297 views

Can compact sets completey determine a topology?

Suppose that $\tau_1$ and $\tau_2$ are two topologies on a set $X$ with the property that $K\subset X$ is compact with respect to $\tau_1$ if and only if $K$ is compact with respect to $\tau_2$. Then ...
0
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2answers
48 views

The image of a compact set under a sequentially continuous real function is bounded

Let $S$ be compact and let $f:S\longrightarrow \mathbf{R}$ be sequentially continuous. Then the image set $f(S)$ is bounded.
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1answer
31 views

Compactness Invariant between normed spaces

Let $X$ and $Y$ be finite dimensional normed spaces. Let $D:\X \rightarrow Y$ be an isometric isomorphism then if $X$ is compact the $Y$ is also compact. I have started by choosing a sequence in $Y$ ...
5
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1answer
77 views

Closure of compact sets in Banach space

Let $(X,\vert\vert\cdot\vert\vert)$ be a Banach space. For each $k\in\mathbb{N}$ let $A_k\subseteq X$ be compact and $r_k\in\mathbb{R},r_k>0$, such that $$A_{k+1}\subseteq \{x+u\vert x\in A_k ...
0
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1answer
26 views

Metric space and compactness

Prove that if in a metric space all closed balls are compact, a subset is compact if and only if it is closed and bounded. Attempt: If all closed balls are compact, then there is a converging ...
3
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1answer
70 views

Limit points and converging subsequences in compact spaces

I need some help to clarify something. I understand that if $X$ is a Hausdorff space but not metric, compact and sequentially compact are not equivalent. This means that there can be sequences ...
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1answer
63 views

Closed subsets of compact sets are compact

If S is a compact subset of R and T is a closed subset of S,then T is compact. (1) Prove this using the definition of compactness. Can somebody prove it? I think we should select a open cover of S ...
1
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1answer
24 views

Is a set of jointly bounded functions over a compact domain compact under p-norm?

Let $X$ be a metric space and a measurable space. Let $K$ be a compact set of nonzero measure and $r> 0$. Is a set $\{ f: K\rightarrow \mathbb R| |f|\leq r$ almost everywhere$\}$ compact with ...
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1answer
84 views

Does sequential compactness imply countable compactness?

Let $X$ be a topological space which is sequentially compact. Does this imply that $X$ is countably compact? Thank you!
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2answers
124 views

theorem compactness and Hausdorff

I have this theorem "$X$ is compact $\leftrightarrow\exp X$ is compact", but i can not find source of it. It concerns Hausdorff metric.
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1answer
28 views

A compactness argument for small high frequencies

I would like to prove the following statement: Let $N\geq 1$, $1\leq q<\infty$ and let be $E$ a relatively compact subset of $L^q(\mathbb{R}^N)$. Then \begin{equation*} \sup_{u\in ...
3
votes
1answer
90 views

Why is $\beta \omega$ compact?

I was trying to show to construct and prove the basic properties of the Stone-Čech compactification of $\omega$ using ultrafilters. I defined $\beta \omega$ as the set of all ultrafilters on $\omega$ ...
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2answers
36 views

Space of probability measures total bounded?

I want to consider a space of probability measures on some set $\Omega$. It's complete (am I right?). But I don't know whether it's total bounded. Actually, I want to prove that the space of ...