Limit point compact

In mathematics, particularly topology, limit point compactness is a certain condition on a topological space which generalizes some features of compactness. In a metric space, limit point compactness, compactness, and sequential 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 a limit point in $X$.

Properties and Examples

* Limit point compactness is equivalent to countable compactness if $X$ is a T₁-space and is equivalent to compactness if $X$ is a metric 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 the countable 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 the indiscrete topology and $mathbb\{Z\}$ is the set of all integers carrying the discrete 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