# Limit point compact

In

mathematics , particularlytopology ,**limit point compactness**is a certain condition on atopological space which generalizes some features of compactness. In ametric space , limit point compactness, compactness, andsequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are mutually inequivalent.A

topological space $X$ is said to be**limit point compact**or**weakly countably compact**if every infinite subset of $X$ has alimit point in $X$.**Properties and Examples*** Limit point compactness is equivalent to

countable compactness if $X$ is a T_{1}-space and is equivalent to compactness if $X$ is ametric space .* An easy example of a space $X$ that is not weakly countably compact is any countable (or larger) set with the

discrete topology . A more interesting example is thecountable complement topology .* Even though a

continuous function from a compact space "X", to an ordered set "Y" in the order topology, must be bounded, the same thing does not hold if "X" is "limit point compact". An example is given by the space $X\; imesmathbb\{Z\}$ (where "X" = {1, 2} carries theindiscrete topology and $mathbb\{Z\}$ is the set of all integers carrying thediscrete topology ) and the function $f=pi\_\{mathbb\{Z$ given by projection onto the second coordinate. Clearly, "f" is continuous and $X\; imes\; mathbb\{Z\}$ is limit point compact (in fact, "every" nonempty subset of $X\; imes\; mathbb\{Z\}$ has a limit point) but "f" is not bounded, and in fact $f(X\; imes\; mathbb\{Z\})=mathbb\{Z\}$ is not even limit point compact.* Every

countably compact space is weakly countably compact, but the converse is not true.* For metrizable spaces, compactness, limit point compactness, and sequential compactness are all equivalent.

* The set of all real numbers is not limit point compact; the integers are an infinite set but do not have a limit point in $mathbb\{R\}$.

*If ("X", "T") and ("X", "T*") are topological spaces with "T*" finer than "T" and ("X", "T*") is limit point compact, then so is ("X", "T")

*A finite space is vacuously limit point compact

**ee also***

Compact space

*Sequential compactness

*Metric space

*Bolzano-Weierstrass theorem *planetmath|id=1234|title=Limit point compact

*planetmath|id=6212|title=Weakly countably compact

*Wikimedia Foundation.
2010.*

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