In mathematics, a topological space X {\displaystyle X} is said to be limit point compact or weakly countably compact if every infinite subset of X {\displaystyle X} has a limit point in X . {\displaystyle 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 {\displaystyle 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 {\displaystyle X} is itself closed in X {\displaystyle X} and discrete, this is equivalent to require that X {\displaystyle X} has a countably infinite closed discrete subspace. Some examples of spaces that are not limit point compact: (1) The set R {\displaystyle \mathbb {R} } of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in R {\displaystyle \mathbb {R} } ; (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 × Y {\displaystyle X=\mathbb {Z} \times Y} where Z {\displaystyle \mathbb {Z} } is the set of all integers with the discrete topology and Y = { 0 , 1 } {\displaystyle Y=\{0,1\}} has the indiscrete topology. The space X {\displaystyle 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 = R , {\displaystyle X=\mathbb {R} ,} the set of all real numbers, with the right order topology, i.e., the topology generated by all intervals ( x , ∞ ) . {\displaystyle (x,\infty ).} The space is limit point compact because given any point a ∈ X , {\displaystyle a\in X,} every x < a {\displaystyle x<a} is a limit point of { a } . {\displaystyle \{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 × Y {\displaystyle X=\mathbb {Z} \times Y} with Z {\displaystyle \mathbb {Z} } discrete and Y {\displaystyle Y} indiscrete as in the example above, the map f = π Z {\displaystyle f=\pi _{\mathbb {Z} }} given by projection onto the first coordinate is continuous, but f ( X ) = Z {\displaystyle f(X)=\mathbb {Z} } is not limit point compact. A limit point compact space need not be pseudocompact. An example is given by the same X = Z × Y {\displaystyle X=\mathbb {Z} \times Y} with Y {\displaystyle Y} indiscrete two-point space and the map f = π Z , {\displaystyle f=\pi _{\mathbb {Z} },} whose image is not bounded in R . {\displaystyle \mathbb {R} .}
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