Limit point 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 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 $$\Reals$$ of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in $$\Reals$$; (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 $$X = \Z \times Y$$ where $$\Z$$ is the set of all integers with the discrete topology and $$Y = \{0,1\}$$ 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 $$X = \Reals,$$ 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 $$x<a$$ is a limit point of $$\{a\}.$$
 * 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 $$X = \Z \times Y$$ with $$\Z$$ discrete and $$Y$$ indiscrete as in the example above, the map $$f = \pi_{\Z}$$ given by projection onto the first coordinate is continuous, but $$f(X) = \Z$$ is not limit point compact.
 * A limit point compact space need not be pseudocompact. An example is given by the same $$X = \Z \times Y$$ with $$Y$$ indiscrete two-point space and the map $$f = \pi_{\Z},$$ whose image is not bounded in $$\Reals.$$
 * 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. Proof: Suppose $$X$$ is a normal space that is not limit point compact. There exists a countably infinite closed discrete subset $$A = \{x_1, x_2, x_3, \ldots\}$$ 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 $$(X, \tau)$$ and $$(X, \sigma)$$ are topological spaces with $$\sigma$$ finer than $$\tau$$ and $$(X, \sigma)$$is limit point compact, then so is $$(X, \tau).$$