# 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.

3,971 questions
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### Characterization of weak compactness in a Banach space

While studying the textbook of Fernando Albiac and Nigel J. Kalton (Topics in Banach Space Theory), I came across the following result: A subset $A$ of a Banach space $X$ is relatively weakly ...
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### Supremum equals maximum on a subset of natural numbers

I'm wondering if $\sup_{x \in M} f(x) = \max_{x \in M} f(x)$ holds when $f$ is some arbitrary function and $M = \{0,1,\dots,n\}$ for some $n \in \mathbb N$. My idea is that $M$ is closed and bounded ...
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### Prove that this operator is compact

Let $E$ be the real Banach space of all real and continuous $\omega$-periodic functions defined on $\mathbb{R}$ with the norm $$\max_{0\leq t\leq\omega}\left | x(t) \right | \:,\:\forall x\in E$$ ...
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### Is there a name for this notion of “radius of compactness” in a metric space?

I was proving some result about Riemannian manifolds that led me to introduce the following definition: Let $M$ be a metric space and $x \in M$. Define the "radius of compactness" $RC(x)$ to be the ...
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### Example of noncompact space in which every real valued continuous function on it is uniformly continuous

I wanted to find Example of non-compact metric space $(X,d)$ such that every real-valued continuous function is uniformly continuous My attempt: $X$ is an infinite set $d$ is a discrete metric. ...
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### On the dimension of rationals over real numbers

The following two are from Introductory Functional Analysis by E Kreyszig: 2.5-2 Lemma . A [sequentially] compact subset M of a metric space is closed and bounded. The converse of this ...
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### Hausdorff, locally connected and locally compact space reference.

I would like to find a reference to the following proposition: Let $X$ a Hausdorff, locally compact, locally connected, connected space and $K \subseteq X$ a compact subset. Then, there exists a ...