Continuous Function is Bounded on Countably Compact Subset

In mathematics, a topological space X is said to be limit point compact [1] [2] or weakly countably compact [3] 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 [edit]

  • 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 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 is itself closed in X and discrete, this is equivalent to require that 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.[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 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 )} .[5] 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.[6]
    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 n : n = 1 , 2 , } {\displaystyle A=\{x_{n}:n=1,2,\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, T) and (X, T*) are topological spaces with T* finer than T and (X, T*) is limit point compact, then so is (X, T).

See also [edit]

  • Compact space
  • Sequentially compact space
  • Countably compact space

Notes [edit]

  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 [edit]

  • James Munkres (1999). Topology (2nd ed.). Prentice Hall. ISBN0-13-181629-2.
  • Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. ISBN 0-486-68735-X (Dover edition).
  • 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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Source: https://en.wikipedia.org/wiki/Limit_point_compact

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