Limit point compact

In mathematics, a topological space [math]\displaystyle{ X }[/math] is said to be limit point compact or weakly countably compact if every infinite subset of [math]\displaystyle{ X }[/math] has a limit point in [math]\displaystyle{ X. }[/math] 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 [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math] is itself closed in [math]\displaystyle{ X }[/math] and discrete, this is equivalent to require that [math]\displaystyle{ X }[/math] has a countably infinite closed discrete subspace.
• Some examples of spaces that are not limit point compact: (1) The set [math]\displaystyle{ \Reals }[/math] of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in [math]\displaystyle{ \Reals }[/math]; (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 [math]\displaystyle{ X = \Z \times Y }[/math] where [math]\displaystyle{ \Z }[/math] is the set of all integers with the discrete topology and [math]\displaystyle{ Y = \{0,1\} }[/math] has the indiscrete topology. The space [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X = \Reals, }[/math] the set of all real numbers, with the right order topology, i.e., the topology generated by all intervals [math]\displaystyle{ (x, \infty). }[/math] The space is limit point compact because given any point [math]\displaystyle{ a \in X, }[/math] every [math]\displaystyle{ x\lt a }[/math] is a limit point of [math]\displaystyle{ \{a\}. }[/math]
• 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 [math]\displaystyle{ X = \Z \times Y }[/math] with [math]\displaystyle{ \Z }[/math] discrete and [math]\displaystyle{ Y }[/math] indiscrete as in the example above, the map [math]\displaystyle{ f = \pi_{\Z} }[/math] given by projection onto the first coordinate is continuous, but [math]\displaystyle{ f(X) = \Z }[/math] is not limit point compact.
• A limit point compact space need not be pseudocompact. An example is given by the same [math]\displaystyle{ X = \Z \times Y }[/math] with [math]\displaystyle{ Y }[/math] indiscrete two-point space and the map [math]\displaystyle{ f = \pi_{\Z}, }[/math] whose image is not bounded in [math]\displaystyle{ \Reals. }[/math]
• 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 [math]\displaystyle{ X }[/math] is a normal space that is not limit point compact. There exists a countably infinite closed discrete subset [math]\displaystyle{ A = \{x_1, x_2, x_3, \ldots\} }[/math] of [math]\displaystyle{ X. }[/math] By the Tietze extension theorem the continuous function [math]\displaystyle{ f }[/math] on [math]\displaystyle{ A }[/math] defined by [math]\displaystyle{ f(x_n) = n }[/math] can be extended to an (unbounded) real-valued continuous function on all of [math]\displaystyle{ X. }[/math] So [math]\displaystyle{ X }[/math] is not pseudocompact.
• Limit point compact spaces have countable extent.
• If [math]\displaystyle{ (X, \tau) }[/math] and [math]\displaystyle{ (X, \sigma) }[/math] are topological spaces with [math]\displaystyle{ \sigma }[/math] finer than [math]\displaystyle{ \tau }[/math] and [math]\displaystyle{ (X, \sigma) }[/math]is limit point compact, then so is [math]\displaystyle{ (X, \tau). }[/math]

See also

• Compact space – Type of mathematical space
• Countably compact space
• Sequentially compact space – Topological space where every sequence has a convergent subsequence.

References

• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. 
• Steen, Lynn Arthur; Seebach, J. Arthur (1995). Counterexamples in topology. New York: Dover Publications. ISBN 0-486-68735-X. OCLC 32311847.