Questions tagged [compactness]

Question 1

I have tried to solve this exercise: Let $H$ be a Hilbert space and let $T = T^* \in B_{\infty}(H)$ a compact operator. Given $\psi_0 \in H$ consider the equations: \begin{align*} (1)\quad &T\psi =...

Question 2

Suppose $X$ is compact Hausdorff and $\mathcal P$ is a partition of $X$. Define the map $\pi:X \to \mathcal P$ taking each point to the unique partition element containing it. Give $\mathcal P$ the ...

Question 3

before I begin, I would like to provide some definitions and theorems. Definition. A topological space $(X, \tau_X)$ is a continuum, if $X$ is a non-empty, metric, compact and connected space. Here $\...

Question 4

I have a question about the proof of the following result in Kashiwara, Schapira, Sheaves on Manifolds: Proposition 2.5.12 [Let $X$ be a Hausdorff and locally compact space.] Let $A$ be a ring, and ...

Question 5

Let $X$ be a Hausdorff topological space and let $\mathcal K$ denote the family of compact subsets of $X$. Assume that the Borel $\sigma$-algebra $\mathcal B(X)$ is generated by compact sets, i.e. $$ \...

Question 6

Let $(M,g)$ be a complete Riemannian manifold and let $d_g:M^2\to\mathbb R$ be the length-minimizing metric induced by $g$. Does it necessarily hold that the compact subsets of $M$ are closed and $d_g$...

Question 7

Given a list of constraints $$ F_i = \{ x\in\mathbb{R}^n\mid L_i(x) \leq c_i \} $$ where $L_i\in L(\mathbb{R}^n,\mathbb{R})$ and $c_i\in\mathbb{R}$ for every $i\in\{ 0,\dots,m \}$ with $m \geq n$, ...

Question 8

I was reading a nice paper, but right at the beginning of one of the main results they claim: $X$ is a locally compact metric space, so it has an equivalent metric such that every closed ball of ...

Question 9

I am trying to prove that if it is true for a set $K$ that every open cover of it has some finite subcovering, then $K$ is sequentially compact and am looking to verify steps that I'm unsure about in ...

Question 10

Let $X,Y$ be Hausdorff and locally compact topological spaces, with $Y$ not compact, and let $\hat{Y}$ be the Alexandroff compactification of $Y$. Let $A\subset X$ be an open, relatively compact ...

Question 11

Let $f:X \rightarrow Y$ be a continuous surjection with $X$ a compact metric space and $Y$ a Hausdorff space. Then $Y$ is metrizable. Does this theorem have a name? Who first proved it? Just ...

Question 12

$\DeclareMathOperator{\int}{int}$Let $(X,(e_\alpha)_{\alpha \in I})$ be a finite cell complex, i.e. $I$ finite. If $X$ is Hausdorff then the closures of the cells $\bar e_\alpha$ are compact hence $X$ ...

Question 13

I am trying to prove the following theorem Theorem - Let $\left( X,d \right)$ be a metric space, the following are equivalent: $X$ is sequentially compact. Every countable open cover $\left\{ U_{n} \...

Question 14

Given a function $f:E\subset\mathbb R^n\longrightarrow\mathbb R$ and $x\in E$. The function $$\omega(f,E):=\sup\{|f(x)-f(y)|:x,y\in E\}$$ is called the oscillation of $f$ on $E$, and $$\omega(f,x):=\...

Question 15

I am currently revising my Topology notes, and today I have started looking into compact metric spaces. I am trying to prove explicitly the following: Theorem - Let $\left( X, d\right)$ be a compact ...

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Questions tagged [compactness]

Compact operators in Hilbert space

Is the quotient of the subspace homeomorphic to the subspace of the quotient?

Order $\omega$ in a dendrite

Sections with compact support commutes with tensor over a locally compact Hausdorff space. Why can we reduce to the compact case?

When compacts generate the Borel $\sigma$-algebra, is $X$ $\sigma$-compact?

Heine-Borel Theorem in Complete Riemannian Manifolds [closed]

When do linear constraints form a compact? [closed]

Uniform Compactness Radius for Locally Compact Metric Spaces

Proving every open cover having finite subcover entails sequential compactness

Exercise on Alexandroff compactification

Attribution for Metrization Theorem

Finite and non-compact (non-Hausdorff) cell complex?

Equivalent conditions for a metric space to be sequentially compact

Oscillation of a function on a compact set with a positive bound

Every compact metric space is complete - without any a priory knowledge of compactness [duplicate]

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