Weakly Countably Compact

In mathematics, 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''. This property generalizes a property of compact spaces. 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 not equivalent.

Properties and examples

* In a topological space, subsets without limit point are exactly those that are closed and discrete in the subspace topology. So a space is limit point compact if and only if all its closed discrete subsets are finite.
* A space ''X'' is ''not'' limit point compact if and only if it has an infinite closed discrete subspace. Since any subset of a closed discrete subset of ''X'' is itself closed in ''X'' and discrete, this is equivalent to require that ''X'' has a countably infinite closed discrete subspace.
* Some examples of spaces that are not limit point compact: (1) The set $\mathbb$ of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in $\mathbb$; (2) an infinite set with the discrete topology; (3) the countable complement topology on an uncountable set.
* Every countably compact space (and hence every compact space) is limit point compact.
* For T₁ spaces, limit point compactness is equivalent to countable compactness.
* An example of limit point compact space that is not countably compact is obtained by "doubling the integers", namely, taking the product $X=\mathbb\times Y$ where $\mathbb$ is the set of all integers with the discrete topology and $Y=\$ has the indiscrete topology. The space $X$ is homeomorphic to the odd-even topology. This space is not T₀. It is limit point compact because every nonempty subset has a limit point.
* An example of T₀ space that is limit point compact and not countably compact is $X=\mathbb$, the set of all real numbers, with the right order topology, i.e., the topology generated by all intervals $(x,\infty)$. The space is limit point compact because given any point $a\in X$, every x is a limit point of $\$.
* For metrizable spaces, compactness, countable compactness, limit point compactness, and sequential compactness are all equivalent.
* Closed subspaces of a limit point compact space are limit point compact.
* The continuous image of a limit point compact space need not be limit point compact. For example, if $X=\mathbb\times Y$ with $\mathbb$ discrete and $Y$ indiscrete as in the example above, the map $f=\pi_$ given by projection onto the first coordinate is continuous, but $f(X)=\mathbb$ is not limit point compact.
* A limit point compact space need not be pseudocompact. An example is given by the same $X=\mathbb\times Y$ with $Y$ indiscrete two-point space and the map $f=\pi_$, whose image is not bounded in $\mathbb$.
* A pseudocompact space need not be limit point compact. An example is given by an uncountable set with the cocountable topology.
* Every normal pseudocompact space is limit point compact.Steen & Seebach, p. 20. What they call "normal" is T₄ in wikipedia's terminology, but it's essentially the same proof as here.
''Proof'': Suppose $X$ is a normal space that is not limit point compact. There exists a countably infinite closed discrete subset $A=\$ of $X$. By the Tietze extension theorem the continuous function $f$ on $A$ defined by $f(x_n)=n$ can be extended to an (unbounded) real-valued continuous function on all of $X$. So $X$ is not pseudocompact.
* Limit point compact spaces have countable extent.
*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'').

See also

*Compact space
*Sequentially compact space
*Countably compact space

Notes

References

*
*Lynn Arthur Steen and J. Arthur Seebach, Jr., ''Counterexamples in Topology''. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).
*{{PlanetMath attribution, id=1234, title=Weakly countably compact

Properties of topological spaces
Compactness (mathematics)