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Mathematics Stack Exchange

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.

6270 questions
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To check whether given set is compact set

why $\prod_{n=1}^{\infty}{A_n}$ with product topology ,where ${A_n=\{0,1\}}$ has discrete topology for n=1,2,3,... is compact set? I know that $\prod_{n=1}^{\infty}{A_n}$ ,where ${A_n=\{0,1\}}$ is uncountable set and is complete as it do not …
dipali mali
  • 555
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(Proof hint)$A = \{0\}\cup\{1, 1/2, ..., 1/n, ...\}$ is Compact

Let $A = \{0\}\cup\{1, 1/2, ..., 1/n, ...\}$ I would like show that A directly satisfies the definition of compactness which is "every open cover has a finite sub-cover" How could one generally select open cover of $A$ to start the proof?
delog
  • 149
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$\{(x,y) \in \Bbb R^2 \mid 0\le x \lt 1, 0\le y \le 1\}$ is not compact

Claim $A = \{(x,y) \in \Bbb R^2 \mid 0\le x \lt 1, 0\le y \le 1\}$ is not compact. I want to prove above claim. I might need to find out finite sub-cover of open cover of given set A. It requires me two step simultaneously, first think about open…
Beverlie
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Let $A = ]0,1]$ Find and open cover with no finite sub-cover

Let $A = ]0,1]$ Find and open cover with no finite sub-cover. I had constructed open cover such as $\{]1/n, 2[ :n = 1,2,3,...\}$ but can't be sure of whether this cover actually contain all of $A$. How could I prove that A actually contain all of…
Daschin
  • 675
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If $A =[ 1/2 ,7/2]$ and $B = (1,9/2)$, show whether $A\cup B$ is compact or not.

If $A =[ 1/2 ,7/2]$ and $B = (1,9/2)$, show whether $A \cup B$ is compact or not. Here $A$ is closed and $B$ is open set , but I not understand that whether $A \cup B$ is closed or open.
Saroj Ghosh
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Must the compact set 's cover be composed only of finite sets?

Im a bit confused with the proof of the fact that K is compact iff K is bounded and closed. More with the - K is compact => is bounded and closed. In my proof i use the fact that If K is compact then for every open cover of K there is a finite…
Zarrie
  • 115
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"if K is compact in R^p, then K x {a} is also compact in R^(p+1)" withough using Heine-Borel Theorem

Should we prove the fact "if K is compact in R^p, then K x {a} is also compact in R^(p+1)" ??? If so, Can We prove "if K is compact in R^p, then K x {a} is also compact in R^(p+1)" by using only definition of compactness ( in other words, by only…
MathHolic
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Is $\{(x,y) \in \mathbb{R}^2 \mid x^3+y^3=1 \}$ compact?

Let $A$ be the subset of points $(x,y) \in \mathbb{R} \times \mathbb{R}$ such that $x^3+y^3=1$. Is $A$ compact? I think it should be, as $x$, $y$ can't have values greater than $1$. So $A$ is bounded. Also it is closed. So, compact. But answer key…
aarbee
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Checking the compactness of sets

I have to check to following sets for compactness in the given spaces with respect to the standard norm for them: \begin{align} M_1 &:= \{f_n:\left[-1, 1\right]\rightarrow \Bbb{R}| f_n(x) = n \cos(nx), n \in \Bbb{N}\} \subset C(\left[-1,1\right])…
vlg
  • 155
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Problem to demonstrate that it is compact

Let $X$ be a Normed Vector Space. For any $x\in X$ and $r>0$, let $W:=\{y∈X:∥y−x∥≤r\}$. Prove: $W$ is closed and if $\dim(X)<\infty$ $W$ is compact. I have no problems show that it is closed, but do not know how to show it is compact. Any…
Peter G
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Surjectiveness in a compact subset

I am completely at a loss as how to proceed. I can't use differentiability here.The question is Let $K$ be a compact subset of $\mathbb{R}$ and $f:K\rightarrow K$ be a function satisfying the condition $|f(x)-f(y)|=|x-y|\ \ \ \ \ \forall \ x,y\in…
AbracaDabra
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Showing Component of Superlevel Set is Compact

Let $f \colon \mathbb{R}^2 \to \mathbb{R}$ be defined by $f(x, y) = \frac{x^2}{\left(x^2 + y^2 + 1\right)^3}$. The superlevel set $$D = \left\{(x,y) \in \mathbb{R}^2 \colon f(x,y) \geq \frac{1}{10}\right\}$$ looks like this and consists of two…
Jon
  • 55
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Show that a set is weakly compact nonseparable.

Suppose that $X$ is a nonseparable weakly compactly generated Banach space. Let $W$ be a weakly compact subset which spans a dense linear subspace of $X$. Denote $\mathcal{F}(X) = \overline{span\{ \delta_x : x \in X \}}$ where $\delta_x$ is an…
Idonknow
  • 15,643
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Show that $K=\{ \frac{x_n}{n} : n \in \mathbb{N} \} \cup \{ 0 \}$ is compact and $X = \overline{span(K)}$

At here, Example $8.2$, there is this statement: Consider any countable and dense subset $\{ x_n : n \in \mathbb{N} \}$ of the unit ball of $X$ and let $K = \{ \frac{x_n}{n} : n \in \mathbb{N} \} \cup \{ 0 \}$. Plainly, $K$ is (weakly) compact and…
Idonknow
  • 15,643
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Compact set in $\mathbb R^n$ with maximum metric

I dont know how to solve this: "Show that $A$ is compact, where $A$ is the set of $x \in\mathbb R^n$ which $\|x\|=1,$ with $\|\cdot\|$ is the maximum metric in $\mathbb R^n.$" My try: I know that in $\mathbb R^n$: a set is compact iff is bounded and…
pipita
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