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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 a collection

Given $\epsilon\in(0,1)$, suppose we have collection $\mathscr{C}(\epsilon)$ of multilinear polynomials in $\Bbb R[x_1,\dots,x_n]$ that on $\{0,1\}^n$ is in range $[-\epsilon,\epsilon]$ on $S_0$ while being in range $[1-\epsilon,1+\epsilon]$ on…
Turbo
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Compact set is nowhere dense in $\mathbb{N}^{\mathbb{N}}$

Show that any compact set is nowhere dense in $\mathbb{N}^{\mathbb{N}}$, the set of all infinite sequences. A set $A$ is nowhere dense if the interior of its closure is empty, i.e. int$(\bar{A})=\emptyset$. I have no idea how to start to show the…
Idonknow
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Prove compact of a set

Could anyone help me to show that the sets $\{(x,y)|f(x,y)\le \gamma, x>0, y>0\}$ are compact for all scalars $\gamma$, for the function $f(x,y)=xy+\frac{1}{x}+\frac{1}{y}$? I think it is easy to show that the set is bounded at $0
lolibility
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Show compactness of $E\cup S_1$

Consider $$ S_1:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\},\\E:=\left\{0\right\}\cup\bigcup_{n\in\mathbb{N}}\left\{(1-2^{-n})e^{\pi i k/2^n}: k\in\left\{0,1,\ldots,2^{n+1}-1\right\}\right\}. $$ Show that $E\cup S_1$ is compact with…
user34632
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Is this a compact metric space?

Consider a fixed set of finite discrete symbols $\mathcal{A}$. Equip $\mathcal{A}$ wit the discrete topology which we denote by $\theta$, and $\mathcal{A}^{\mathbb{Z}^d}$ with the product topology, denoted by $\tau$. Is then…
Salamo
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prove using the definition of compactness

If A is compact and B is closed then A Intersection B is compact. I tried to solve it using the fact that compact set is closed and bounded but the problem here they did not tell us in which topology is compact or if its compact in R.
Yahya Ram
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How to prove the sequentially compact of a set of sequences.

anyone can gives me some clues on how to solve this problem? I think there is something need to do with the 1/j. But I just have no idea about how to prove this. Any suggestions appreciated! Thanks so much.
lalala8797
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How could we state that $A$ and $B$ are compact if $A\times B$ is compact in $M \times N$?

Assume that the Cartesian product of two non-empty sets $A \subseteq M$ and $B\subseteq N$ is compact in $M \times N$. Which steps should I follow to prove that $A$ and $B$ are compact?
seyfullah
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compact and closed set problem

Let {Ωα : α ∈ I} be an arbitrary family of closed sets Ωα ⊆R^d with an index set I. (a) Prove that ⋂Ωα is a closed set. [7] (b) Set d = 2 and show by construction of a counterexample that ⋃ Ωα is not necessarily closed. [2] I understand the…
J.floyd
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An operator is not compact

let $T$ be the operator on $ \ell_{2}$ (the complex Hilbert space of square summable sequences), defined by $$ T(x_{1},x_{2},x_{3},\dots)=(0,x_{2},x_{3},\dots).$$ Show that $T$ is not compact.
A. Bag
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Could you help me : compactness of a set from compact product

If S is compact and S= $A\times B$ product of two spaces. Is that enough to state that continuous image of a compact space is compact to state A is compact if we define $f: S\rightarrow$ A is continuous ?
seyfullah
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If $A, B$ are two open sets in $\mathbb R$ such that $A\cap B $ is compact, then show ....

If $A, B$ are two open sets in $\mathbb R$ such that $A\cap B$ is compact, then show that $A\cap B =\varnothing$. Since A is open then int$(A)\subseteq A$, and same thing for $B$, but I can't solve this.
Saroj Ghosh
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