• 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 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). ==See also==