Limit point compact

In mathematics, a topological space X {\displaystyle X} is said to be limit point compact[1][2] or weakly countably compact[2] if every infinite subset of  X {\displaystyle X} has a limit point in  X . {\displaystyle 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

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* In a topological space, subsets without limit point are exactly those that are closed and discrete in the subspace topology. So a space is limit point compact if and only if all its closed discrete subsets are finite.
* A space  X {\displaystyle X} is not limit point compact if and only if it has an infinite closed discrete subspace. Since any subset of a closed discrete subset of  X {\displaystyle X} is itself closed in  X {\displaystyle X} and discrete, this is equivalent to require that  X {\displaystyle X} has a countably infinite closed discrete subspace.
* Some examples of spaces that are not limit point compact: (1) The set  R {\displaystyle \mathbb {R} } of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in  R {\displaystyle \mathbb {R} } ; (2) an infinite set with the discrete topology; (3) the countable complement topology on an uncountable set.
* Every countably compact space (and hence every compact space) is limit point compact.
* For T1 spaces, limit point compactness is equivalent to countable compactness.
* An example of limit point compact space that is not countably compact is obtained by "doubling the integers", namely, taking the product  X = Z × Y {\displaystyle X=\mathbb {Z} \times Y} where  Z {\displaystyle \mathbb {Z} } is the set of all integers with the discrete topology and  Y = { 0 , 1 } {\displaystyle Y=\\{0,1\\}} has the indiscrete topology. The space  X {\displaystyle X} is homeomorphic to the odd-even topology.[3] This space is not T0. It is limit point compact because every nonempty subset has a limit point.
* An example of T0 space that is limit point compact and not countably compact is  X = R , {\displaystyle X=\mathbb {R} ,} the set of all real numbers, with the right order topology, i.e., the topology generated by all intervals  ( x , ∞ ) . {\displaystyle (x,\infty ).} [4] The space is limit point compact because given any point  a ∈ X , {\displaystyle a\in X,} every  x < a {\displaystyle x<a} is a limit point of  { a } . {\displaystyle \\{a\\}.}
* For metrizable spaces, compactness, countable compactness, limit point compactness, and sequential compactness are all equivalent.
* Closed subspaces of a limit point compact space are limit point compact.
* The continuous image of a limit point compact space need not be limit point compact. For example, if  X = Z × Y {\displaystyle X=\mathbb {Z} \times Y} with  Z {\displaystyle \mathbb {Z} } discrete and  Y {\displaystyle Y} indiscrete as in the example above, the map  f = π Z {\displaystyle f=\pi _{\mathbb {Z} }} given by projection onto the first coordinate is continuous, but  f ( X ) = Z {\displaystyle f(X)=\mathbb {Z} } is not limit point compact.
* A limit point compact space need not be pseudocompact. An example is given by the same  X = Z × Y {\displaystyle X=\mathbb {Z} \times Y} with  Y {\displaystyle Y} indiscrete two-point space and the map  f = π Z , {\displaystyle f=\pi _{\mathbb {Z} },} whose image is not bounded in  R . {\displaystyle \mathbb {R} .}
* A pseudocompact space need not be limit point compact. An example is given by an uncountable set with the cocountable topology.
* Every normal pseudocompact space is limit point compact.[5]
Proof: Suppose  X {\displaystyle X} is a normal space that is not limit point compact. There exists a countably infinite closed discrete subset  A = { x 1 , x 2 , x 3 , … } {\displaystyle A=\\{x_{1},x_{2},x_{3},\ldots \\}} of  X . {\displaystyle X.} By the Tietze extension theorem the continuous function  f {\displaystyle f} on  A {\displaystyle A} defined by  f ( x n ) = n {\displaystyle f(x_{n})=n} can be extended to an (unbounded) real-valued continuous function on all of  X . {\displaystyle X.} So  X {\displaystyle X} is not pseudocompact.
* Limit point compact spaces have countable extent.
* If  ( X , τ ) {\displaystyle (X,\tau )} and  ( X , σ ) {\displaystyle (X,\sigma )} are topological spaces with  σ {\displaystyle \sigma } finer than  τ {\displaystyle \tau } and  ( X , σ ) {\displaystyle (X,\sigma )} is limit point compact, then so is  ( X , τ ) . {\displaystyle (X,\tau ).}

## See also

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* Compact space – Type of mathematical space
* Countably compact space
* Sequentially compact space – Topological space where every sequence has a convergent subsequence

## Notes

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1. ^ The terminology "limit point compact" appears in a topology textbook by James Munkres where he says that historically such spaces had been called just "compact" and what we now call compact spaces were called "bicompact". There was then a shift in terminology with bicompact spaces being called just "compact" and no generally accepted name for the first concept, some calling it "Fréchet compactness", others the "Bolzano-Weierstrass property". He says he invented the term "limit point compact" to have something at least descriptive of the property. Munkres, p. 178-179.
2. ^ a b Steen & Seebach, p. 19
3. ^ Steen & Seebach, Example 6
4. ^ Steen & Seebach, Example 50
5. ^ Steen & Seebach, p. 20. What they call "normal" is T4 in wikipedia's terminology, but it's essentially the same proof as here.

## References

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* Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. (accessible to patrons with print disabilities)
* Steen, Lynn Arthur; Seebach, J. Arthur (1995) [First published 1978 by Springer-Verlag, New York]. Counterexamples in topology. New York: Dover Publications. ISBN 0-486-68735-X. OCLC 32311847.
* This article incorporates material from Weakly countably compact on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.