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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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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68 views

Is Baire space $\sigma$-compact?

Is Baire space $\sigma$-compact? Thank you!
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1answer
42 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
51 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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92 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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78 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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99 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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50 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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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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60 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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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
75 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
38 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
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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385 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
64 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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52 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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83 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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90 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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186 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
89 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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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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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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76 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
45 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
59 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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403 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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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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242 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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486 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
112 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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94 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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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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105 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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34 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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423 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
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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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80 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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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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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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246 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 ...
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61 views

A counterexample on compactness (closed vs complete)

In a metric space $M$: If $A \subset M$ is complete and for each $\epsilon > 0$ there exists a compact $K \subset M$ with $A \subset \{ x \in M : d_M(x, K) \leq \epsilon \}$ then $A$ is compact. ...
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19 views

About Weakly Lindelöf Determined Banach spaces

I'd like to know where can I read more about weakly lindelöf determined (WLD) spaces. Especifically, I need to prove: 1.- Every weakly compactly generated is WLD 2.- If X is WLD then (X*,w*) is ...
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68 views

Locally-compact function spaces?

I ask this question out of curiosity, not a specific need. Euclidean spaces and manifolds. Are there examples of locally compact function spaces? Could (some?) Sobolev spaces be locally compact?
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62 views

Čech-Stone compactification of $\mathbb N$ and ultrafilters on $\mathbb N$

I have found in the literature that the Čech-Stone compactification $\beta\mathbb N$ of $\mathbb N$ (or more generally, of any discrete topological space) can be identified with ultrafilters on ...