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.
6,438 questions
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Ahlfors' complex analysis proof doubt - Heine-Borel theorem
I am currently reading Ahlfors' complex analysis. I am confused by his proof that a complete, totally bounded metric space $S$ satisfies the Heine-Borel property (by Heine-Borel property we mean any ...
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Does sequential compactness guarantee every convergent subsequence in S converges to a point in S
Let $E,F \subseteq \mathbb{R}^d$ be compact sets. Define
$$ d(E,F) := \inf_{x \in E, y \in F} |x - y|$$
Prove that there exists points $\hat{x} \in E,\hat{y} \in F$ such that
$$ d(E,F) = |\hat{x} - \...
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Continuity of point to set mapping
A point-to-set map $\Omega(x): X \subseteq \mathbb{R}^n \rightrightarrows Y \subseteq \mathbb{R}^p$ is called
open at a point $\bar{x} \in X$ if for all sequences $\{ x^k \} \subseteq X $ with $x^k ...
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Proof $X=\Bbb R/\Bbb Z$ is not homeomorphic to $B$.
I want to show that the space $B$ is not homeomorphic to the space $X$, where
$$
B:=\bigcup_{n \in \mathbb{Z}^{+}} C_n=\left\{(x, y) \in \mathbb{R}^2 \left\lvert\,\left(x-\frac{1}{n}\right)^2+y^2=\...
- 864
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Is $\omega_1$ Meta-Lindelöf?
A space is meta-Lindelöf provided every open cover of $X$ has a point-countable open refinement. (A collection $\mathscr V$ of subsets of $X$ is point-countable if each point $x\in X$ belongs to at ...
- 8,011
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Same topologies using Alaoglu's Theorem
I am trying to prove that for $0 < p < r< 1$ the weak-* topologies $\tau_p:=\sigma(l_{\infty}, l_p)$ and $\tau_r:=\sigma(l_{\infty}, l_r)$ are the same when restricted to norm-bounded subsets ...
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Is it possible for the feasible regions of an LP problem and its dual to both be compact?
I'm sure this is easy for experts, but I'm not sure how to see it myself and I can't find an answer in any of the references I skimmed through.
We know that if a (maximizing) linear program is ...
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Is there a metacompact space that fails to be para-Lindelöf?
A space is metacompact provided every open cover of the space has a point-finite open refinement.
A space is para-Lindelöf (resp. paracompact) provided every open cover of the space has a locally ...
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Show that for all $f\in C(X,\mathbb{R})$ there's a sequence $(f_n)_{n\in\mathbb{N}}$ of $\mathcal{F}$ that converges pointwise to $f$.
Let $X$ be a Hausdorff $\sigma$-compact space and $\mathcal{F}\subseteq C(X,\mathbb{R})$ be a subalgebra that separates points of $X$. Show that for all continuous functions $f\in C(X,\mathbb{R})$ ...
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If A is closed then $X/A$ is a hausdorff space [duplicate]
I m trying to proof the following statement:
Let $X$ be a topological space and $A\subset X$ a subset. We write $X/A$ as de quotient space of $X/\sim$ wherein $\forall x,y \in X: x \sim y \iff (x =y $ ...
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Proving compactness is needed to assure the Lebesgue Number Lemma holds
I am attempting to prove compactness is needed to assure the Lebesgue Number Lemma holds via an actual example. My thought is using the interval $\left[0,1\right]$ then comparing that to $\left(0,1\...
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Prove that on a Hausdorff $\sigma$-compact space every Baire measure is tight
Let $X$ be a topological space. We say that an open set $U\subseteq X$ is functionally open if there are a continuous function $f:X\to \mathbb{R}$ and an open set $V\subseteq\mathbb{R}$ such that $U=f^...
- 1,341
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Is there a "maximal" compact, totally disconnected Hausdorff space with a countable base of clopen subsets?
It is a well-established fact that the Cantor set $\mathcal C$ is the perfect, compact, totally disconnected Hausdorff space with a countable base of clopen subsets.
But, more broadly, is $\mathcal C$ ...
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Compactness of Set of Monotone Functions of Probability Measures
I would like to prove compactness of the following set:
$$\mathcal{F}= \left\{f\colon\Re_+ \times \mathcal{P}(\Re_+) \longrightarrow [0,1] : f(\cdot , \mu) \ \text{increasing, for every} \ \mu \in \...
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Questions about S. Solecki, Analytic ideals and their applications.
I read from "S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic, 99 (1999),51–71."
In the first; I can't understand what is the analytic ideal? I know what the ideal ...
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Why is Hawaiian Earring Sequentially Compact?
I am not looking for a really formal proof, but I remember the reasoning was based on the fact that a subsequence can be found on one of the circles. However I cannot justify this and I don't have any ...
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Discontinuous function on R that can be uniformly approximated by polynomials
This is my first question here, I have read from here that any almost everywhere continuous function that is almost everywhere equal to a continuous function, can be uniformly approximated by ...
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Question about coverings of zero Hausdorff measure compact sets (crossposted from mathoverflow)
Consider a compact set $E\subset \mathbb{R}^n$ ad assume that ${\cal H}^{n-1}(E)=0$.
If $\epsilon>0$, there is a finite covering of $E$ made of open balls $B_{r_k}(x_k)$ such that $\sum_{k}r_k^{n-1}...
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Are continuous real valued functions on a compact group uniformly continuous? [duplicate]
Suppose $G$ is a compact topological group and $f\colon G\to \mathbb R$ a continuous function. From some bachelor course in analysis I have seen a result stating that continuous functions from a ...
user1457233
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Function different from the null function on $L^2(\mathbb{R})$ with compact support whose Fourier transform also has compact support.
Problem: "Decide if there exists a function different from the null function on $L^2(\mathbb{R})$ with compact support whose Fourier transform also has compact support. If it exists, give an ...
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Show a set is compact or not on $C[0, 1]$
Let $$S = \left\{ x_n(t) = \frac{1}{1 + t^n} : n \in \mathbb{N} \right\}$$ Determine whether $ S $ is compact in the space $C[0, 1]$
If I understand the task correctly, $C[0,1]$ is the space of all ...
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What Topologies Can Be Realized Using a Closed Base of Compact Sets
Given an arbitrary topological space $(X,\tau)$, note that the union of two compact sets is always compact, although the intersection of compact sets need not be compact.
So we can create a new ...
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Are $\sigma$-localy compact sets exhaustible by compact sets?
Claim: Any countable union of compact subsets of a Euclidean space is exhaustible by compact sets.
Proof: If $F$ is a subset of a Euclidean space $X$, it is locally compact, because the Euclidean ...
- 1,544
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Generalization of compactness and local compactness
Consider the operation $r_X(A) = \overline{A}\setminus A$ in a space $X$. Here, suppose that $X$ is a compact Hausdorff space.
Call a space $A$ to be $n$-compact in $X$, if $r_X^n(A) = \emptyset$ ...
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Closed cells of a CW-complex are compact [closed]
Are closed cells of a $CW$-complex always compact?
In my definition, a $CW$-complex is a Hausdorff space $X$ with a partition (the cells) with for each cell $C$ a map $\phi : D^n \to C$ ($n$ is the &...
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Does every door space with an infinite compact set have a single non-isolated point?
A door space has the property that every subset is either closed or open.
Note that discrete spaces are door spaces (every subset is both closed and open).
A simple non-discrete space that is a door ...
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Compactness and closed sets
I'm struggling with the proof (in the book Analisi Matematica 1 Pagani-Salsa) of the Heine-Borel theorem, the theorem states:
Given $E \subset \mathbb{R}^n$, then $E$ is compact $\iff$ $E$ is closed ...
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Compactness of a space of sequences
I consider the set $X=\{ (x_n)_{n\in\mathbb{N}} : \forall n\geq 0, x_n\geq 0,\;\sum_{n\geq 0}x_{n}\leq 1 \}$. I would like to show that $(X,d)$ where
$$
d(x,y) = \sum_{n\geq 0}\beta^n\lvert x_n - y_n\...
- 1,641
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Does every product of copies of $\mathbb Z$ fail to be pseudocompact?
A space is pseudocompact if every continuous image of the space into $\mathbb R$ is bounded.
Clearly $\mathbb Z\subseteq\mathbb R$ is not pseudocompact, as inclusion is in unbounded continuous mapping....
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Compact set of Irrational number is nowhere dense
I am trying to show that:
Let $(\mathbb{R} \setminus \mathbb{Q}, \tau_d)$ be the irrational number space under the Euclidean topology $\tau_d$.
Let $S \subseteq \mathbb{R} \setminus \mathbb{Q}$ be a ...
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If every totally bounded and closed set is compact, is the space complete?
I know a well-known result that if every closed and bounded set is compact, then the whole space is complete. I wonder if this is true when a stricter condition is imposed, that is, only every totally ...
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Addition map is continuous under compact convergence topology
Let $X$ be a topological space. Endow $\mathcal C(X,\mathbf R)$ with the compact convergence topology.
Prove: The addition, multiplication and the scalar multiplication
$$\begin{aligned}a:\mathcal C(...
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Compactness of unit sphere in 1-norm; Gamelin and Greene
In Introduction to Topology by Gamelin and Greene, I'm working an exercise to show the equivalence of norms in $\mathbb R^n$. This exercise succeeds another exercise where various equivalent ...
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Does $A$ weakly compact in $\mathrm{C}(E,\mathbb{C})$, with $E$ compact, implies $A$ is compact?
Let $E$ be a compact Hausdorff space, and consider the Banach space $\mathrm{C}(E,\mathbb{C})$, of the continuous complex-valued functions, with the supremum norm.
Does $A$ weakly compact in $\mathrm{...
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Is every second-countable weakly locally compact space also locally compact?
A second countable space has a countable basis. A weakly locally compact space has a compact neighborhood for each point of the space; in particular, every compact space is weakly locally compact. A ...
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Class of compact sets is compact class
Class $\mathfrak{K}$ of subsets of X is compact class if for every countable $\{K_n\}$, $K_n \in \mathfrak{K}$, next conditions are equivalent:
\begin{align*}
\bigcap_{n=1}^{\infty}K_n=\emptyset \iff \...
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Doubt on compact space proof
The topological space $(X,\mathcal{T})$ is compact and exists a compact subset $K$ that is not closed. I want to see that
$$
\mathcal{T}_{K^c} = \{ U \cup ( V \cap K^c) \: : \: U,V \in \mathcal{T} \}
$...
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Proving a point preserving compactness of subsets is a limit point
Let B be an infinite set in $\mathbb{R}$. (Countable or uncountable - we are not given any more details).
Let $p \in \mathbb{R}$ be a point such that for every subset $C \subseteq B$ we have $C \cup \{...
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If $K$ compact, then $\cup_{x \in K} \overline{B_r(x)}=A$ is compact
In class they gave the following exercise, which I haven't been able to prove.
It reads: In $\mathbb{R}^n$, given $K$ compact and $r>0$, then $A=\cup_{x \in K}\overline{B_r(x)}$ is also compact.
...
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Compact Resolvent Operator for Schrodinger operator
I encounter problems with the compact operator shown in $L^{\infty}$.
Let $\rho \in C^{\infty}_c(-L,L)$, I want to first show $\rho R_0(\lambda)$, where $R_0(\lambda) = (-\Delta + \lambda^2)^{-1}$, is ...
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Exercise 6.5 of Atiyah and Macdonald
A topological space $X$ is said to be Noetherian if the open subsets of $X$ satisfy the ascending chain condition (or, equivalently, the maximal condition). Since closed subsets are complements of ...
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Understanding some concepts regarding metric spaces. [closed]
I already know the base definition and 4 axioms. The things I don't know are completeness, compactness, and connectivity.
Also, in the example:
$\gamma: [0, 1]\to D$, $\gamma(0) = z_1, \gamma(1) = z_2$...
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Well-separated minimum on compact set
On the simplex $\Omega = (w_1,...,w_K : 0 \leq w_k, \sum_k w_k = 1)$ (I'm unsure why the curly brace isn't rendering), I have a continuous function $S: \Omega \rightarrow \mathbb{R}$, and I assume ...
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Disjoint union of compact Hausdorff spaces is compact Hausdorff
Consider a compact Hausdorff space $A$. Suppose we have a family of compact Hausdorff spaces
$\{B_a\}_{a\in A}$ indexed by the space $A$. Then is it true that the space $$X=\sqcup_{a\in A} B_a$$ is ...
- 5,700
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Is a set defined by a bounded gradient compact? (with log-barrier function)
Consider an augmented function $f$ consisting of (the original cost) function $f_{0}$ with log-barrier functions $f_{i}, i \in \{1, 2\}$ such that
$$
f(x) = f_{0}(x) - \frac{1}{c}\sum_{i=1}^{2} \log(-...
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Can two compact Hausdorff spaces have the same convergent sequences?
In a lecture that I attended a few days ago, the lecturer seemed to indicate that given a compact Hausdorff space $(X,\tau)$, if we know exactly which sequences in $X$ that are convergent and if we ...
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Proof of continuous image of compact set is compact
Theorem: Let $K\subset \mathbb{R}^n$ be compact and $f:K\to \mathbb{R}^m$ continuous. It holds that $f(K)$ is compact.
I know the proof is contained in Spivak's Calculus on manifolds, however, I ...
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How to debunk a proof that compact space is sequentially compact [duplicate]
A recent comment under this old answer claims there's some issue with the following proof.
Prop. A compact metric space $(M,d)$ is sequentially compact.
The "proof":
Suppose a sequence $(...
- 2,030
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Is the converse of the Tube Lemma true? If $\pi\colon X\times Z \to Z$ is always closed, is $X$ compact?
One of the forms of the Tube Lemma is the following: if $X$ is compact, then for every topological space $Z$, the projection mapping $\pi_Z: X\times Z \to Z$ is closed.
In various algebraic geometry ...
- 2,689
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Compact mapping of differential operators
Suppose we have functions $F : \mathbb{R}^n \to \mathbb{R}$ that must satisfy some differential equation on $\mathbb{R}^n \to \mathbb{R}$. Additional we write $\mathbb{R}^n$ as the typical spherical ...
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