Limit point compact
In mathematics, a topological space X is said to be limit point compact 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 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 and the function given by projection onto the second coordinate. Clearly, ƒ is continuous and is limit point compact but ƒ is not bounded, and in fact is not even limit point compact.
- Every countably 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 and are topological spaces with T* finer than T and is limit point compact, then so is.
- A finite space is vacuously limit point compact.
