Limit point compact

In mathematics, a topological space ${\displaystyle X}$ is said to be limit point compact ^{[1] [2]} or weakly countably compact ^[3] if every infinite subset of ${\displaystyle X}$ has a limit point in ${\displaystyle X.}$ 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

• 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 ${\displaystyle X}$ 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 ${\displaystyle X}$ is itself closed in ${\displaystyle X}$ and discrete, this is equivalent to require that ${\displaystyle X}$ has a countably infinite closed discrete subspace.
• Some examples of spaces that are not limit point compact: (1) The set ${\displaystyle \mathbb {R} }$ of all real numbers with its usual topology, since the integers are an infinite set but do not have a limit point in ${\displaystyle \mathbb {R} }$; (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 ${\displaystyle X=\mathbb {Z} \times Y}$ where ${\displaystyle \mathbb {Z} }$ is the set of all integers with the discrete topology and ${\displaystyle Y=\{0,1\}}$ has the indiscrete topology. The space ${\displaystyle X}$ is homeomorphic to the odd-even topology. ^[4] 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 ${\displaystyle X=\mathbb {R} ,}$ the set of all real numbers, with the right order topology, i.e., the topology generated by all intervals ${\displaystyle (x,\infty ).}$ ^[5] The space is limit point compact because given any point ${\displaystyle a\in X,}$ every ${\displaystyle x<a}$ is a limit point of ${\displaystyle \{a\}.}$
• 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 ${\displaystyle X=\mathbb {Z} \times Y}$ with ${\displaystyle \mathbb {Z} }$ discrete and ${\displaystyle Y}$ indiscrete as in the example above, the map ${\displaystyle f=\pi _{\mathbb {Z} }}$ given by projection onto the first coordinate is continuous, but ${\displaystyle f(X)=\mathbb {Z} }$ is not limit point compact.
• A limit point compact space need not be pseudocompact. An example is given by the same ${\displaystyle X=\mathbb {Z} \times Y}$ with ${\displaystyle Y}$ indiscrete two-point space and the map ${\displaystyle f=\pi _{\mathbb {Z} },}$ whose image is not bounded in ${\displaystyle \mathbb {R} .}$
• 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. ^[6]
  Proof: Suppose ${\displaystyle X}$ is a normal space that is not limit point compact. There exists a countably infinite closed discrete subset ${\displaystyle A=\{x_{1},x_{2},x_{3},\ldots \}}$ of ${\displaystyle X.}$ By the Tietze extension theorem the continuous function ${\displaystyle f}$ on ${\displaystyle A}$ defined by ${\displaystyle f(x_{n})=n}$ can be extended to an (unbounded) real-valued continuous function on all of ${\displaystyle X.}$ So ${\displaystyle X}$ is not pseudocompact.
• Limit point compact spaces have countable extent.
• If ${\displaystyle (X,\tau )}$ and ${\displaystyle (X,\sigma )}$ are topological spaces with ${\displaystyle \sigma }$ finer than ${\displaystyle \tau }$ and ${\displaystyle (X,\sigma )}$is limit point compact, then so is ${\displaystyle (X,\tau ).}$

See also

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

Notes

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. Steen & Seebach, p. 19
3. Steen & Seebach, p. 19
4. Steen & Seebach, Example 6
5. Steen & Seebach, Example 50
6. Steen & Seebach, p. 20. What they call "normal" is T₄ in wikipedia's terminology, but it's essentially the same proof as here.

References

• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN   978-0-13-181629-9. OCLC   42683260.
• 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.