“Limit point compact”的意思、由来-开放百科全书

词条

Limit point compact

释义

Properties and Examples
See also
Notes
References

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 (where X = {1, 2} carries the indiscrete topology and is the set of all integers carrying the discrete topology) and the function given by projection onto the second coordinate. Clearly, ƒ is continuous and is limit point compact (in fact, every nonempty subset of has a limit point) but ƒ is not bounded, and in fact 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.
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.

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

{{cite book

| author = James Munkres
| year = 1999
| title = Topology
| edition = 2nd
| publisher = Prentice Hall
| isbn = 0-13-181629-2
}}

{{PlanetMath attribution|id=1234|title=Weakly countably compact}}

2 : Properties of topological spaces|Compactness (mathematics)

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

Properties and Examples

See also

Notes

References