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Questions tagged [compactness]

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. This includes logical compactness.

Compactness is a topological property. We say that a topological space $X$ is compact if whenever we cover $X$ by a collection of open sets we can find a finite number of open sets from the collection which cover $X$. For example, $[0,1]$ is a compact subspace of $\mathbb{R}$, but $(0,1)$ and $\mathbb{R}$ are not.

We say that a space $X$ is sequentially compact if every sequence has a convergent subsequence. These properties are equivalent for metric space, although neither implies the other in general.

This tag may also be used for questions about logical compactness, such as the compactness theorem.

More information can be found on Wikipedia.

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Compactness of subset of $\mathbb{R}$

The following is a problem from my analysis homework: Let $A$ be an infinite set in $\mathbb{R}$ with a single accumulation point in $A$. Must $A$ be compact? What I'm having trouble understanding is the hypothesis. How can an infinite subset of…
Kevin Sheng
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More about compact.

Any advice to solve the following problem? Let $K$ be a compact set of the real numbers. Prove that the set $|K|=\{|x|: x\in K\}$ is a compact set. Thanks a lot!
EQJ
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Compactly contained subset

Let $E\subset\mathbb{R}^n$ be a bounded domain (open and connected) and $F\Subset E$ be a compactly contained (or even compact) subset of $E$. We set $a := dist(F,\partial E) = \inf\limits_{ y\in F} \inf\limits_{x\in \partial E}|x-y|>0$, where the…
HelloEveryone
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A 1-1 continuous function from a compact space A into a Hausdorff space B

I am following a proof for the theorem: Let g be a 1-1 continuous function from a compact space $A$ into a Hausdorff space $B$. Then $A$ and $g[A]$ are homeomorphic. Proof:Clearly $g:A \rightarrow g[A]$ is onto, and given that $g$ is 1-1 and…
Vinod
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Compactness of Gaussian Random Field

When watching a video on youtube, an instructor says that Gaussian Random Field (GRF) is not compact. I understand that in GRF, we draw sample from function $u(x)$ from, in the case of zero mean $$\mathcal{N} \sim (0, K), \quad K_{i,j} =…
user1168149
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Bolzano-Weierstrass property think of it conversely

I am reading this Bolzano-Weierstrass property saying that a metric space X is sequentially compact if every sequence in X has a convergent subsequence. I also read a theorem saying that sequentially compact is equivalent to compact. So I am…
Halk
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Alternative proof for compactness under the inverse of a closed mapping

I encountered the following problem in L. Gasínski, N.S. Papageorgiou - Exercises in Analysis - Part 1: (2.63) "Suppose that $X$ and $Y$ are two topological spaces and $f : X \longrightarrow Y$ is a closed function such that for every $y \in Y$ ,…
Werner
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How do I find assumptions on M such that M gets compact?

I have a problem understanding the meaning of this question: Consider a set $M$ endowed with the discrete, the indiscrete or the cofinite topology, under which assumptions on M does this yeld a compact space? As I understood this, I first endow…
user123234
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What is Compactness and why is it useful?

I would like to gain a better understanding of the notion of compactness (topology). The wiki definiton defines a compactness of an interval as closed and bounded. In mathematics, specifically general topology, compactness is a property that…
SheppLogan
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Intersection of a family of compact sets in Hausdorff topological space is compact

Let $\{G_i\}_{i \in \Lambda}$ be a family of compact sets in a Hausdorff topological space. Show that the intersection $\bigcap_{i \in \Lambda} G_i$ is compact. My attempt was: Since $G_j$ for some $j \in \Lambda$ is compact, there exists an open,…
user866761
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Compact sets are closed.

I am a beginner in studying functional analysis, so please pardon me if my argument seems absurd. I am having hard time, coming in terms with the proof of "Compact subsets of $\mathbb{R}$, are closed". Now lets say that I have a set $ A = [1,2) $,…
Prashanth
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Definitions of relative compactness

Per definition, a subset $A$ of a metric space $(X,d)$ is relatively compact if and only if its closure $\overline{A}$ is compact. That means that any sequence in $\overline{A}$ contains a subsequence that converges in $\overline{A}$. Does that…
user77614
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Using an open and compact set to find another compact set within the original compact set.

I am trying to show that if U a subset of R^n is open, and K a subset of U is compact, then there is a compact set such that K is a subset of D^o and D is a subset of U. I know that since K is compact and K is a subset of U, we can use the boundary…
acme_2020
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Compact set in $\mathbb{R}^n$ and the finite intersection property

Show that a set $K ⊂ \mathbb{R}^n$ is compact if and only if every family of closed subsets of K having the finite intersection property has nonempty intersection. I was able to prove one side (That if $K$ is compact then every family of closed…
Math101
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$\{z: |z|\leq R\}$ is compact ($z$ is a complex number, Proof verification)

Here's my proof: for each $z=x+iy=(x,y)$ ($x,y$ are real numbers), $|z|=|x^2+y^2|\leq R$ so, $\{z:|z|\leq R\}$ is equal to $\{(x,y):|x^2+y^2|\leq R\}$ then, by Heine-Borel Theorem in $\mathbb{R}^2$, $\{z:|z|\leq R\}$ is compact. Is it correct? (i…
Choco Mint
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