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 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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58 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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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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113 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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918 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
89 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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67 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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333 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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231 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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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
127 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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90 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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136 views

Compactness and closedness

If every closed and proper subset of a topological space $X$ is compact, then is the whole space necessarily compact? The "converse" of this question is well-known, of course, but I'm having ...
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98 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
125 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
72 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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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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94 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
43 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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Difference between closed, bounded and compact sets

In real analysis, there is a theorem that a bounded sequence has a convergent subsequence. Also, the limit lies in the same set as the elements of the sequence, if the set is closed. Then when ...
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514 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
205 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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199 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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136 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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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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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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83 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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120 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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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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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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329 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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1answer
85 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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1answer
145 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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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 ...
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1answer
44 views

How can a bounded subspace of the left order topology be compact?

I want to show that every bounded set equipped with the left order topology is compact. This is a statement I found on a wikipedia page and appearently it is lifted from the book Counterexamples in ...
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108 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
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1answer
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Analysis question.

Is this set compact? $\{(x,y) \in \mathbb R^2 : |x|+|y|\leq 1\}$. I know that is closed and bounded so compact but I don't know how to show it is closed and bounded mathematically. This is the graph ...
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1answer
62 views

Compact features

Consider this problem: Let $X$ be a metric space, $U$ be open, $K$ compact and $K\subset U$, show that there exists a $r>0$ such that $B(k,r)\subset U$ $\forall k\in K$ Here $B(k,r)=\{x\in X ...
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351 views

Proof help. Core-compactness, Hausdorff, Locally Compact

While reading about topologies on continuous function spaces, I've seen remarks that core-compact and locally compact are equivalent for Hausdorff spaces. Now I can clearly see that locally compact ...
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4answers
221 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
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Compactness in topology of uniform conergence (of functions and all their derivatives) on compact subsets of (0,\infty)

I am trying to understand an example in the book "Lectures on Choquet's Theorem" (R.R. Phelps). My question is: Given the space of real valued infinitely differentiable functions on $(0, \infty)$ ...
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1answer
166 views

Sequence of elements having a convergent subsequence -NBHM $2014$

Question is to find which of the following are true? Let $V$ be the space of continuous functions on $\mathbb{R}$ with compact support endowed with metric ...
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1answer
70 views

Noncompactness of the closed unit ball in $L^2$

Let $$ L^2[0,1]=\{f:[0,1]\to\mathbb R\,\,\text{such that}\,\, \|f\|_2<∞\}, $$ where $\|f\|_2^2=\int_0^1 |f(x)|^2\,dx.$ Show that the unit sphere $$ S=\{f\in L^2[0,1]:\|f\|_2\le 1\}, $$ is ...
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Connectedness and compactness of a union of two sets

Let: $$A=\Big\{ (x,y) \in \mathbb R^2: 0 \le x \le 1, y=\frac{x-1}{n},\, n\in \mathbb N \Big\}$$ $$B=\Big\{ (x,y) \in \Bbb R^2: 0 \le x \le 1, y=\frac{x}{n},\, n\in \mathbb N \Big\}$$ Is $A \cup B$ ...
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1answer
51 views

One-point compactification of the union of circle and an intersecting open interval.

I have to give the one-point compactification of $S^1 \cup \{(0,2) \times \{0\}\}$. I think I can see this as a circle with two 'tails' with open ends, one on the inside, one on the outside. Is this ...
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$\sigma$-$\sigma$-compactness is $\sigma$-compactness?

I mean, if $X=\displaystyle\bigcup_{n\in\mathbb{N}}K_n$ where each $K_n$ is $\sigma$-compact, then $X$ is $\sigma$-compact? I'm not sure if a countable union of countable unions is still a countable ...
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83 views

Is the following set is compact

Consider the set of all $n \times n$ matrices with determinant equal to one in the space of $\mathbb R^{n\times n}$. My idea is compact because determinant function is continous ant it is bijective ...
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1answer
38 views

Show that $A$ is non-compact

I have a problem: For $C\left [ 0,1 \right ]=\left \{ x:\left [ 0,1 \right ] \to \Bbb R \ \text{is continuous on } \left [ 0,1 \right ] \right \}$, with a norm: $$\left \| x \right \|=\sup_{t\in ...
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Is an ideal generated by a compact subset finitely generated?

Let $R$ be a commutative topological ring and let $K$ be a compact subset of $R$. Denote by $I$ the ideal generated by $R$. Then is it true (or under what assumptions on $R$ (besides Noethernity)) is ...