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

• Limit point compactness is equivalent to countable compactness if X is a T₁-space and is equivalent to compactness if X is a metric space.
• An example of a space X that is not weakly countably compact is any countable (or larger) set with the discrete topology. A more interesting example is the countable complement topology.
• 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 ${\displaystyle X\times \mathbb {Z} }$ (where X = {1, 2} carries the indiscrete topology and ${\displaystyle \mathbb {Z} }$ is the set of all integers carrying the discrete topology) and the function ${\displaystyle f=\pi _{\mathbb {Z} }}$ given by projection onto the second coordinate. Clearly, ƒ is continuous and ${\displaystyle X\times \mathbb {Z} }$ is limit point compact (in fact, every nonempty subset of ${\displaystyle X\times \mathbb {Z} }$ has a limit point) but ƒ is not bounded, and in fact ${\displaystyle f(X\times \mathbb {Z} )=\mathbb {Z} }$ is not even limit point compact. ${\displaystyle X\times \mathbb {Z} }$ is an example of limit point compact space that is not countably compact.
• Every countably compact space (and hence every compact space) is weakly countably compact, but the converse is not true.
• For metrizable spaces, compactness, limit point compactness, and sequential compactness are all equivalent.
• 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.

See also

• Compact space
• Sequential compactness
• Metric space
• Bolzano-Weierstrass theorem
• Countably compact space

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.

References

• James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.
• This article incorporates material from Weakly countably compact on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.