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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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Heine-Borel clarification

Text I am reading states: Let $S$ be a compact set. Suppose $p \notin S$. For each point $q \in S$, consider an open neighborhood $W_q$ of radius less than half $d(p, q)$. Such a system of open neighborhoods covers $S$ because $q \in W_q$ for…
LeastSquaresWonderer
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Bounded and Closed but not Compact

I had been proposed to construct not compact set that is however bounded and closed. I could easily imagine from the different metric - such as discrete metric where $d(x,y) =0$ if $x=y$ and $d(x,y) =1$ if $x \neq y$ then M itself is closed since…
Daschin
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Tips on determining compact to non-compact sets

I find it hard to quickly determine this. For instance, $D^n/\{0\}=\{x \in \mathbb{R}^n| 0 < ||x|| \leq 1\}$ is non-compact. $S^{n-1}=\{x \in \mathbb{R}^n | ||x||=1\}$ is compact. So, okay, the first set is a punctured disc, it seems. So a disc,…
John Trail
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Show that the unit sphere with centre $0$ in $\mathbb{R}^d$ is compact.

Namely, the sphere is $\{x\in\mathbb{R}^d: \| x\|_2=1\}$. I am going about this by proving that the sphere is bounded and closed. I have proved that it is bounded and I can see that it must be closed but I don't know how to write it out, can this be…
user187039
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Finiteness of a compact subset in $\mathbb R^n$

Let $K$ be a compact subset of $\mathbb R^n$ such that for all $x \in K$, $K\setminus\{x\}$ is also compact. Show that $K$ is finite. I'm trying to solve it using sequences, but am having difficulty. Could someone help me? Thank you!
user137153
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Kernel and Compactness

Let $A \subseteq \mathbb{R}^n$ and let the kernel of $A$ be: $Ker(A) = \{x \in A\, |\,[x,y] \subset A, \forall y \in A\}$. I want to show that if $A$ is compact, then $Ker(A)$ is compact. I even do not know where to start and how to start. Any help…
Hypatia
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largets level set

Consider the set $$S=\{x,y,z|2x^2+y^2+z^2+xz<5\}$$ how can we find the largest level set of $S$ on $x-y$ plain? basically I look for a compact set $\bar{S}(x,y)$ such that every $x,y\in S$ is also in $\bar{S}$. Is the following correct? the max of…
Eric Brown
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What is the largest possible compact set that can fit inside an open set?

If $U$ is open and $C \subset U$ is compact, there is a compact set $D \subset U$ such that $C \subset \text{interior}(D)$. Now, by inductive reasoning, this means that there is actually a chain (maybe infinite?) of compact sets $C, D, E, F, .....$…
Awe Kumar Jha
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Prove that monotone sequence in $K$ that does not converges to a point in $K$

Suppose that $K$ is a non-empty subset of the set of real numbers and is non compact. Prove that there exists a monotone sequence in $K$ that does not converge to a point in $K$.
김기필
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Partition on metric space

Let be $\gamma = \{C_1,\ldots, C_k\}$ a partition of a compact metric space $X$ such that $diam(C_j)<\delta$ for all $j$. Suppose that there exist compact sets $L_i\subset C_i$ for all $i\in\{1,\ldots,k\}$ and pairwise disjoint. I want to prove that…
Teo Viddoto
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compactness proof in Munkres

In Munkres, he proves that every closed subspace Y of a compact space X is compact. In the proof, he adjoins the open set X - Y to an open cover of Y, and then he goes on to prove Y is compact. Why did he adjoin X - Y to this open cover? Could he…
Britt K
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Relatively compact and compact set exercise

I have che following exercise and some dubts: Let $M>0$ and $\mathcal{F}=\{ f\in C^{1}([a,b]) \, | \, \| f \|_{C^{1}} \leq M\}$. Prove that $\mathcal{F}$ is relatively compact in $(C^{0}([a,b]), \| \cdot \|_{\infty})$ and $\mathcal{F}$ is not…
Giovanni
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Let X be sequentially compact. Then X is also countably compact.

A subset $A $ of a topological space $X$ is sequentially compact iff every sequence in $A $ has a convergent subsequence. Countably compact means every countable open cover has a finite subcover. I don't understand how can I link these two…
Soumyadweep Mondal
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Compactness implies countably compactness

Let X be compact. Then X is countably compact. My thinking is like this: Let X be a topological space. Since compact, then every open cover hava a finite subcover. Hence it is true for countable open cover. Is this proof true?
Soumyadweep Mondal
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Compactness of a set given by two equalities

Let be $v_1<\cdots
byk7
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