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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.

6270 questions
9
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Does a cover have to be infinite

I have encountered this question: find a cover for $[1,2]$ in $\mathbb{R}$ that does not have a finite subcover. Since $[1,2]$ is closed and bounded then it's compact. It then implies that a cover that does not have a finite subcover must not be…
user1691278
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5
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3 answers

Show this set is closed

As part of a proof I am writing for analysis, I need to show the following set is closed: $F_n = \{x \in \mathbb{R} \, | \,x \ge 0, ~~ 2-\frac{1}{n} \le x^2 \le 2+\frac{1}{n}\}$. My current approach is to show that $F_n = \overline{F_n}$. I.e. I…
Kevin Sheng
  • 2,483
5
votes
1 answer

Property of compact, convex sets in $\mathbb{R}^3$

How to solve the following: Let $K\subset \mathbb{R}^3$ be a convex, compact set with smooth boundary $C=\partial K$ and let $\vec{u}$ be any vector. Show that there exist points $x\neq y$, $x,y\in C$ such that vector $\vec{xy}$ is colinear with…
alans
  • 6,475
5
votes
2 answers

Tychonoff's theorem for $[0,1]^\mathbb{R}$

According to Tychonoff's theorem any uncountable product of compact spaces is compact with respect to product topology. Then $[0,1]^\mathbb{R}$, the space of all functions defined on $\mathbb{R}$ taking values in $[0,1]$ is compact w.r.t. the…
Seyhmus Güngören
  • 7,848
4
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0 answers

Is $ \left\{\cup _{n \in \mathbb{N} } (x, x^n) \mid x \in [0,1] \right\} $ compact?

Is $$ \left\{\cup _{n \in \mathbb{N} } (x, x^n) \mid x \in [0,1] \right\} $$ compact? My answer would be it isn't because it doesn't contain all its accumulation points ( for example the point $ (1/2,0) $)
user15269
  • 1,632
4
votes
5 answers

Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact

Show that the closed unit ball $B[0,1]$ in $C[0,1]$ is not compact under the following metrics: $1. d(f,g)=\sup_{x\in [0,1]}|f(x)-g(x)|$ $2.d(f,g)=\int _0^1 |f(x)-g(x)| dx$ My try: In order to show not compact if we can find a sequence which has no…
Learnmore
  • 31,062
3
votes
2 answers

Proof that Compact set is Closed and Bounded

I came across the following problem while reading Spivak's Calculus on Manifolds. Prove that a compact subset of $\mathbb{R}^n$ is closed and bounded. The definition of an open set in the book is a set $A$ in which for all $x \in \mathbb{A}$,…
Tae Hyung Kim
  • 718
3
votes
1 answer

Which of the following are compact?

Which of the following are compact? $1$. The set of all upper triangular matrices all of whose eigenvalues satisfy $|\lambda|\leq 2$. $2$. The set of all real symmetric matrices all of whose eigenvalues satisfy $|\lambda|\leq 2$. $3$. The set of…
Learnmore
  • 31,062
2
votes
2 answers

Counter example of a locally compact topological space which is not compact

I want to show that not every locally compact topological space is compact. I have one example which I am not sure if it is a correct example which is simply the whole real line. Is it a correct example? Can you provide more examples of such…
Avrham Aton
  • 445
2
votes
1 answer

Compact subset inbetween another compact subset

Assume we have $A\subset \mathbb{R}^n$ open (regarding standard topology). If we have $B\subset A$ compact with dist$(B,\partial A)=\epsilon >0$, can we find $C\subset A$ compact with $B\subset C \subset A$ and dist$(B,\partial A)= \delta >0$ such…
HelloEveryone
  • 449
2
votes
0 answers

Let $A$ be a compact subset of $R^{n}$ and let $B$ be an open set containing $A$.

Let $A$ be a compact subset of $\mathbb{R}^{n}$ and let $B$ be an open set containing $A$. Prove that there exists a $C^{\infty}$ function $h$ on $\mathbb{R}^{n}$ such that $h=1$ on $A, h=0$ outside $B$, and $0 \leq h \leq 1$ in $B-A$. I have an…
Bảo Ngọc Hoàng
  • 21
2
votes
0 answers

Compactness of the set of functions from a compact set to a finite set

Let $(K,\mathcal{B},P)$ be a probability space where $K$ is a compact set, $\mathcal{B}$ is the Borel sigma-algebra and P is a Borel probability measure. Let $A$ be an arbitrary finite set. Consider the space of measurable functions from $K$ to…
FraGarb
  • 49
2
votes
2 answers

Is $\{0,1\}^{\mathbb{N}}$ compact in $\mathbb{N}^{\mathbb{N}}$?

Is $\{0,1\}^{\mathbb{N}}$ compact in $\mathbb{N}^{\mathbb{N}}$ ? How can i proof that?
user679342
2
votes
1 answer

Does the given set is compact in $\mathbb{R^{2}}$?

Check whether given set on $\mathbb{R^{2}}$ is compact or not ? $ \left\{(x,y) \in \mathbb{R^{2}}\,\middle|\,x>0,y=\sin\left(\frac{1}{x}\right)\right\}\bigcap\left\{(x,y) \in \mathbb{R^{2}}\,\middle|\,x>0,y=\frac{1}{x}\right\}$ First i have to…
gaurav saini
  • 763
2
votes
3 answers

About sequential compactness

If $A\subset \mathbb R^n$ is compact, $x\in A$, and let $\{x_i\}$ be a sequence in $A$ such that every convergent subsequence of $\{x_i\}$ converge to $x$. Prove that $\{x_i\}$ also converge. How to even approach this problem, by contradiction?
JFK
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