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# weakly countably compact

A topological space $X$ is said to be *weakly countably compact*
(or *limit point compact*)
if every infinite subset of $X$ has a limit point.

Every countably compact space is weakly countably compact. The converse is true in $\mathrm{T}_{1}$ spaces.

A metric space is weakly countably compact if and only if it is compact.

An easy example of a space $X$ that is not weakly countably compact is any infinite set with the discrete topology. A more interesting example is the countable complement topology on an uncountable set.

Defines:

limit point compactness, weak countable compactness

Keywords:

topology

Related:

Compact, CountablyCompact, SequentiallyCompact, PseudocompactSpace

Synonym:

limit point compact, limit-point compact

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

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54D30*no label found*

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new question: Prove a formula is part of the Gentzen System by LadyAnne

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new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

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new correction: Misspelled name by DavidSteinsaltz

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new correction: underline-typo by Filipe

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new correction: cocycle pro cocyle by pahio

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new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier