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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25 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
85 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
32 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
51 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
63 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
67 views

Is Baire space $\sigma$-compact?

Is Baire space $\sigma$-compact? Thank you!
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1answer
40 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
40 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
48 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
37 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
84 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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0answers
69 views

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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2answers
98 views

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
43 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
68 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 ...
2
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1answer
59 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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3answers
50 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
64 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
35 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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3answers
53 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
98 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
364 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
63 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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2answers
50 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
77 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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87 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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180 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
84 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
70 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. $A$ ...
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2answers
108 views

Compactness and closedness

If every closed and proper subset of a topological space is compact, then is the whole space necessarily compact? The "converse" of this question is well-known, of course, but I'm having difficulty ...
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3answers
73 views

Compact Set Question

Consider the topology $\tau$ defined on $\mathbb{R}$ by $U\in\tau$ iff $\forall s\in U$, $\exists t>s$ such that$[s,t)\subseteq U$. Show that $[0,1]$ is not compact. My attempt: We only need to ...
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1answer
43 views

Is my understanding of limit point compactness correct with respect to $[0,1]^{\omega}$ with the uniform topology?

The following is an exercise problem about limit point compactness from the book "Topology" by Munkres (2nd edition). Exercise 1 in Section 28: Give $[0,1]^{\omega}$ the uniform topology. Find an ...
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1answer
54 views

A theorem on compactness

A result about compactness says that A topological space is compact if every basic open cover has a finite subcover. The proof runs as follows: Let$\{G_i\}$ be an open cover and $\{B_j\}$ an open ...
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3answers
374 views

Inverse image of a compact set is compact

Let $X$ and $Y$ be topological spaces, $X$ compact, $f : X \to Y$ continuous. Then the preimage of each compact subset of $Y$ is compact. With the stipulation that $X$ and $Y$ are metric spaces, this ...
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0answers
70 views

US does not imply AB

We say that a topological space $X$ is: $AB$, provided that $X$ is $T_1$ and for each pair $(A, B)$ of compact, disjoint subsets of $X$ there is $U$, an open subset of $X$, such that either $A ...
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1answer
30 views

If $X$ is not countably compact, then there exists a countable subset without accumulation points

I want to prove that if $X$ is not countably compact, then there exists a countable subset $\{x_n:n\in\mathbb{N}\}$ and has no accumulation points. If $X$ is not CC, then there exists an open cover ...
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2answers
203 views

Difference between closed, bounded and compact sets. [closed]

Can somebody explain the difference between compact, bounded and closed sets with examples?
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483 views

What's wrong with this 'open cover' of the Koch Snowflake?

This question is to help me find peace. First, the question of the Snowflake's compactness has been tackled here on this site: Is the Koch Snowflake a Compact Space? Is Koch snowflake a continuous ...
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1answer
110 views

Connected and Compact preserving function is not continuous example?

Before we start, I'm aware the result is true for when the function is a map between Euclidean spaces. In fact, with a minimal amount of extra work we can see that a function between locally-compact, ...
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0answers
91 views

Counterexample to Converse of Extreme Value Theorem?

The extreme value theorem says: If $X$ is a compact topological space, then for all functions $f: X \to \mathbb{R}$ such that $f$ is continuous we have that $f$ satisfies the extreme value property. ...
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29 views

Compactness in Sobolev spaces

I am looking for characterizations of compactness in the Sobolev space $H^{-1}$. In particular, I am looking for a characterization involving the Fourier transform. Can anyone suggest some results ...
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2answers
104 views

a compact set $X$ has a countable set $S$ such that $\overline{S} = X$

Suppose $X \subseteq \mathbb{R}^d$. Suppose $X$ is compact. Then there exists a countable subset of $X$, $S \subseteq X$ such that $\overline{S} = X$. How can I show this? I have no idea how to ...
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0answers
33 views

Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
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389 views

If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
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2answers
73 views

Proving the set $C = \{\,x \in \mathbb R^n : \sum x_i = 1, x_i \in [0,1]\,\}$ is compact.

Proving the set $C = \{\,x \in \mathbb R^n : \sum_{1}^n x_i = 1, x_i \in [0,1]\,\} \subseteq \mathbb R^n$ is compact. Alright: I can use the Heine-Borel theorem to prove this, therefore all I need to ...
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1answer
94 views

Arbitrary union of compact sets in a topological space $X$ is not necessarily compact [closed]

Is the countable set $\{1/n | n\in\mathbb{N}\}\cup\{0\}\subset\mathbb{R}$ compact with the usual topology its arbitrary union is compact or not?
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2answers
77 views

Compact Domain and Inverse Image

I am trying to show that given $f:M \rightarrow N$, where $M$ is compact, $f$ is continuous and onto, then given $A \subset N$: $$ f^{-1}(A) \text{ closed} \implies A\text{ closed} $$ I am dealing ...
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1answer
66 views

Open, closed, bounded or sequentially compact

How do I find if this set is open, closed, bounded or sequentially compact? $$S=\left\{z:5<\left|z\right|\leq7\right\}$$ I find the value of $z$ is: $-7\le z < 0$. Can you please explain. Thank ...
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0answers
48 views

Equivalence conditions in the Heine-Borel theorem for the real line

The Heine Borel theorem (book, pg 335) shows that the following conditions are equivalent- A set $K$ is closed and bounded. $K$ is compact. My question is that in the proof of 1 $\implies$ 2 where ...
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2answers
243 views

Compactness implies Continuity?

I am stuck on this question (probably there are many counterexamples, but I can't find any). "Suppose $f:\mathbb{R}\mapsto\mathbb{R}$ that preserves compactness (i.e, for every $K \subseteq R$, then ...