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.

• 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 ).}

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

• 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.