Limit point compact explained

X is said to be limit point compact^{[1] [2]} or weakly countably compact^[3] 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 x 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.^[4] 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, inf ty). ^[5] 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 x 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 x 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.^[6]
  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).

References

• • Book: Steen. Lynn Arthur. Lynn Arthur Steen. Seebach. J. Arthur. J. Arthur Seebach Jr.. Counterexamples in topology. Dover Publications. New York. 1995. First published 1978 by Springer-Verlag, New York. 0-486-68735-X. 32311847.

Notes and References

1. The terminology "limit point compact" appears in a topology textbook by James Munkres where he says that historically such spaces had been called just "compact" and what we now call compact spaces were called "bicompact". There was then a shift in terminology with bicompact spaces being called just "compact" and no generally accepted name for the first concept, some calling it "Fréchet compactness", others the "Bolzano-Weierstrass property". He says he invented the term "limit point compact" to have something at least descriptive of the property. Munkres, p. 178-179.
2. Steen & Seebach, p. 19
3. Steen & Seebach, p. 19
4. Steen & Seebach, Example 6
5. Steen & Seebach, Example 50
6. Steen & Seebach, p. 20. What they call "normal" is T₄ in wikipedia's terminology, but it's essentially the same proof as here.