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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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continuous maps compact set to compact set.

If $A$ is a subset of $\Bbb R$ and $f : A \to \Bbb R$ is a continuous function then prove or disprove the following. If $A$ is a bounded set but not closed then $f(A)$ is bounded. If $A$ is a closed set but not bounded then $f(A)$ is closed. My…
Baljeet
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Show closedness , path connectedness and compactness

Let f : R2 →R be a continuous function. Let S = {(x,y,z): z = f(x,y)}. Show that S is closed, path connected but not compact. Unfortunately inspite of knowing the definitions, I donot know how to prove
Didi
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Are these sets bounded and compact or not?

I am interested in whether the following 2 sets are open, closed, bounded, and compact. 1) $A = \bigl\{(x, y) \in \mathbb{R^2}\quad|\quad|x − y| > 4\bigr\}$ 2) $B = \bigl\{(x, y, z) \in \mathbb{R^3}\quad|\quad x + y − z \le 1, x^2 + y^2 + z^2 \ge…
user656813
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How can I tell from the equation $x^3+y^3=\frac{1}{\sqrt{2}}$ that it is an unbounded set of values

$x^3+y^3=\frac{1}{\sqrt{2}}$ is according to my textbook not bounded since, for each $x_0 \in R$ there is a $y_0$ such that the point $(x,y_0)$ is on the curve. Isn't that true for a bounded set also e.g $x^2+y^2=1$. Im not sure what this statement…
F Wi
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proof compactness of sets

Let $$K ⊂ R^m$$ and $$L ⊂ R^n$$ be compact subsets. Show: a) The set $$ K × L: = \{(x, y): x ∈ K, y ∈ L\} ⊂ R^{m+n}$$ is also compact. I have to show that this is bounded and closed. But how do I do that?
breakshooter
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I don't even understand this question (compactness)

Suppose that $E$ is a compact subset of $\mathbb R$. If for every $x\in E$ there exist a nonnegative function $f=f_x$ and an $r=r(x)>0$ such that $f$ is $C^\infty$ on $\mathbb R$, $f(t)=1$ for $t\in (x-r, x+r)$, and $f(t)=0$ for $t\notin…
JFK
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On the question of lp compactness. Counterxamples

Is it a unity norm, closed sphere, in lp, (not Lp ), compact ? I hear that it is not !! Is there a counterexample for this or a proof ? Thanks a lot !
EL ANDREAKIS
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Function $0$ after a certain t is equivalent to one defined on a compact set?

Let's define the following set of functions: $$C_c(\mathbb{R}) = \{f \in C(\mathbb{R}) | f(t) = 0, \forall t \text{ s.t. } |t| \geq T, T \geq 0\} $$ Then, can I say that for a general $f \in C_c(\mathbb{R}) $, $f$ is defined in $[-T,T]$ ? Because…
ofir_13
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Variation of Tietze extension theorem

Let $X$ be a compact Hausdorff space and let $A \subseteq X$ be closed. By Tietze extension theorem for any function $f\in C(A, \mathbb{C})$ we find a map $F\in C(X, \mathbb{C})$ such that $F|_A=f$. Is it possible to choose $F$ with $F(X)=f(A)$?
worldreporter
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Definition of Compact : Is "Sub"-cover Must be Proper?

Here's a given definition while I am reading the Topology, Munkres. Definition. A space $X$ is said to be compact if every open covering $A$ of $X$ contains a finite subcollection that also covers $X$. Then my question here, when we check whether…
Beverlie
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Compact covering of a compact set.

This question comes from Advanced Calculus second edition by Patrick M. Fitzpatrick. For each natural number $n$, let $I_n$ be a closed bounded interval. Suppose that $\{I_n\}_{n=1}^\infty$ covers the compact set consisting of the closed bounded…
Walt
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Spivak proof partition of unity: case 2 (countable compact sets)

I have a question about Spivak's proof on the partition of unity (see below). Spivak first proves the case where $A$ is compact. Then he proceeds with case 2: I don't really understand what this $U$ here is. Is $U$ simply an open subset of…
Sha Vuklia
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Understanding the proof that $X$ is compact.

Definition of compact that I'm using: $E$ is compact if each of its open covers has a finite subcover. Example: Let $X = \mathbb{R}$ with the topology $T = \left \{ U \subset X: U = X \text{ or } 1 \notin U \right \}$. We will show $X$ is compact.…
Dominated Convergence Theorem
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every $k$ cell is compact

I did not understand the proof from the word "and some sets in this infinite sequence are from $\{G\}$....." to the end. Thanks for helping me to understand. $k$ cell in $\mathbb{R}^k$ is defined as $\{x=(x_1,\dots, x_k):a_i\le x_i\le…
Myshkin
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Prove that $(1/n,2/n)$ is an open cover of $(0,1)$

Let $F$ the collection of the open intervals $(1/n,2/n)$, $n\geq 2$. Show that $F$ is an open cover of $(0,1)$. I can get $n,m\in\mathbb{R}$ such that if $0
John
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