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.

• 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 ${\displaystyle (X,\tau ).}$

• Compact space – Type of mathematical space
• Countably compact space – topological space in which from every countable open cover of the space, a finite cover can be extractedPages displaying wikidata descriptions as a fallback
• Sequentially compact space – Topological space where every sequence has a convergent subsequence

• 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.