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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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 ...
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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 ...
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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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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
31 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
105 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: ...
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1answer
71 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
58 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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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 ...
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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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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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299 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 ...
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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 $Y$ ...
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1answer
78 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 ...
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1answer
27 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 ...
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1answer
78 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
66 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 ...
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1answer
25 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
89 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
125 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
32 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 ...
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1answer
91 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 ...
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1answer
58 views

Strong convergence of bounded sequences in Bochner spaces

Let $S=(0,T)$ for a $T>0$ and let $B_0,\ B_1,\ B_2$ be Banach spaces, such that $B_0$ is compactly embedded in $B_1$, which is in turn continuously embedded in $B_2$. Suppose we have a sequence ...
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1answer
75 views

A generalization of the generalized tube lemma

I am trying to prove the following generalization of the generalized tube lemma: Let $\{X_t\}_{t \in T}$ be a family of Hausdorff spaces and $\prod_{t \in T}A_t$ be a compact subset of $X=\prod_{t ...
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1answer
68 views

Is Baire space $\sigma$-compact?

Is Baire space $\sigma$-compact? Thank you!
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1answer
44 views

Topological contraction on compact spaces

This is a follow up question. You can see the original here. I have the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a ...
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1answer
41 views

What is wrong with my argument? (copy of $\beta\omega$ in $\mathbb R$)

Let $(a_n)$ be a strictly increasing sequence in $[0,1]$. Then {$a_n:n\in\omega$} is relatively discrete in $[0,1]$. So $cl_{[0,1]}${$a_n:n\in\omega$}$\simeq ...
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1answer
56 views

What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
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1answer
38 views

Is there any Banach space $X$ that $L^2(\Omega)$ is compactly embedded into?

Let $\Omega \subset \mathbb{R}^n$. Is there a good (*) Banach space $X$ that $L^2(\Omega)$ is compactly embedded into: $$L^2(\Omega) \subset\!\subset X$$? If not compactly embedding, I at least would ...
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1answer
98 views

Equivalent definitions of Compact Sets?

Usually, compact sets are defined by each open cover of the set having a finite subcover. My professor gave us a bizarre definition: A set X is said to be compact if each infinite subset has an ...
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Measures with bounded total variation norm compact in $M(X)$?

Let $X$ be a separable, metric, compact space. (e.g. an interval in $\mathbb{R}$ like $[0,10]$). Let $M(X)$ be the set of all finite signed measures over $X$ with weak-*-topology (in probability ...
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Contraction of compact sets

I am trying to solve the following problem. Let $X$ be a compact Hausdorff space and let $f:X\to X$ be continuous. Show that there exists a non-empty set $A\subset X$ such that $f(A)=A$. ...
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1answer
59 views

Compactness and connectedness of the topological space?

Let $X=\mathbb N$ be equipped with the topology generated by the basis consisting of sets $A_n = \{n,n+1,n+2,\ldots\} ,n \in \mathbb N $ . Then $X$ is compact and connected Hausdorff and connected ...
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1answer
69 views

Whether a space is compact, if all functions are bounded

Let $X$ be a paracompact Hausdorff space. It is easy to see the following statement. If $X$ is compact, then every continuous function is bounded. Does the converse hold? If every function on ...
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1answer
62 views

A boolean algebra is complete if its stone space is extremally disconnected

I have the following proof, but I don't understand one of the steps: Theorem 4.4. A Boolean algebra is complete iff its Stone space is exlremally disconnected. Proof. Identify the given ...
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52 views

Find a locally compact space $X$ with a subspace $A$ that is NOT locally compact.

I'd like to find a locally compact space $X$ with a subspace $A$ that is NOT locally compact. As from here, I know that if $A$ is closed and $X$ is Hausdorff, then $A$ is locally compact. Anyone ...
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1answer
83 views

A closed subspace of a locally compact Hausdorff space is also a locally compact Hausdorff space.

Let $X$ be a locally compact Hausdorff space, and $A$ a closed subspace. Show that $A$ is a locally compact Hausdorff space. Here is what I have for a proof. Will I need to clarify anything else? ...
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1answer
39 views

Closed subspace of a compact topological space is compact

Let $X$ be a compact topological space, and $A$ a closed subspace. Show that $A$ is compact. How does this look? Proof: In order to show that $A$ is compact. We need to show that for any open ...
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57 views

Understanding the definition of compactness.

$(X, \mathscr T )$ be a topological space and $A \subset X$. $\{ U_i \mid i \in I \}$ is said to be an open cover of $A$ if $A \subset \cup_{i \in I} U_i$. $A$ is said to be compact if there exists ...
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1answer
99 views

Confusion on Compact Space

Definition. If for any open cover $\mathcal U$ of $X$, there exists a finite subcover $\mathcal V$ of $\mathcal U$, we call $X$ is compact. Theorem 1. Let $X$ be compact. If $\{F_n\}_{n\in\mathbb ...
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1answer
393 views

What is the one-point compactification of $\mathbb{Z}_{+}$?

The problem arises from the exercise 29.8 of the book "Topology" by Munkres: Show that the one-point compactification of $\mathbb{Z}_{+}$ is homeomorphic with the subspace $\{ 0 \} \cup \{ 1/n ...
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1answer
66 views

Prove a that a topological space is compact iff

Prove that the topological space $X$ is compact $\Leftrightarrow$ whenever {$C_j:j\in J$} is a collection of closed sets with $\bigcap_{j\in J}C_j = \varnothing$, there is a finite subcollection ...
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54 views

compact in the product topology

I am going to check if $\{f \in X: |f(t)|<1 \text{ for all } t \in[0,1]\}$ is compact in the product topology $X = \mathbb R^{[0,1]}$. I suspect that this would not be compact since it may not be ...
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1answer
93 views

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed

Prove that if sets $A$ and $B$ are closed and bounded then $A+B$ is closed I know that $A$ and $B$ are closed and bounded, then they are sequentially compact, so $A+B$ also sequentially compact, ...
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98 views

The continuous image of a sequentially compact set is also sequentially compact.

Let $S$ be a sequentially compact set and let $f : S\to R$ be continuous. Then the image $f(S)$ is sequentially compact.
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187 views

A Theorem About Compactness and

My first exposure to any sort of topology is from Spivak's Calculus on Manifolds. I think I understand compactness conceptually, I'm just finding the rigor a little bit elusive. My first question ...
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1answer
90 views

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact.

Show that if $G$ is a locally compact topological group and $H$ is a subgroup, then $G/H$ is locally compact. This seems pretty straight forward but how will I be able to prove this? I saw this ...
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3answers
73 views

On compact sets

Let $A$ be a subset of $\mathbb R$ with more than one element. Let $a\in A$. If $A\setminus \{a\}$ is compact, then $A$ is compact. every subset of $A$ must be compact. $A$ must be a finite set. $A$ ...