# 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 T_{1}-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.

## See also

## Notes

- 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.*

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