Limit point compact

In mathematics, a topological space X is said to be limit point compact[1] 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

• Even though a continuous function from a compact space X, to an ordered set Y in the order topology, must be bounded, the same thing does not hold if X is limit point compact. An example is given by the space $X\times\mathbb{Z}$ (where X = {1, 2} carries the indiscrete topology and $\mathbb{Z}$ is the set of all integers carrying the discrete topology) and the function $f=\pi_{\mathbb{Z}}$ given by projection onto the second coordinate. Clearly, ƒ is continuous and $X\times \mathbb{Z}$ is limit point compact (in fact, every nonempty subset of $X\times \mathbb{Z}$ has a limit point) but ƒ is not bounded, and in fact $f(X\times \mathbb{Z})=\mathbb{Z}$ is not even limit point compact.
• Every countably compact space (and hence every compact space) is weakly countably compact, but the converse is not true.
• The set of all real numbers, R, is not limit point compact; the integers are an infinite set but do not have a limit point in R.
• If (X, T) and (X, T*) are topological spaces with T* finer than T and (X, T*) is limit point compact, then so is (X, T).
• A finite space is vacuously limit point compact.

Notes

1. ^ The terminology "limit point compact" appears in a topology textbook by James Munkres, and is apparently due to him. According to him, some call the property "Fréchet compactness", while others call it the "Bolzano-Weierstrass property". Munkres, p. 178–179.