Countable compact space

In the mathematical subfield of topology , countable compactness denotes a weakening of the concept of compactness, which is central to the theory of topological spaces .

definition

A topological space is called countably compact if every countable open cover of has a finite partial cover . A subset of a topological space is countably compact if and only if it is countably compact as a topological space with the subspace topology . ${\displaystyle (X,\tau )}$ ${\displaystyle X}$ ${\displaystyle A\subseteq X}$

properties

Of course, every countably compact Lindelöf room is also compact and every compact topological room is also countably compact.

A topological space is countably compact if and only if every filter on which has a countable filter base is contained in a convergent filter. ${\displaystyle (X,\tau )}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X}$

Every sequentially compact topological space is countably compact. Conversely , if a countable compact topological space fulfills the first countability axiom , then it is compact in terms of consequences.

A topological space is countably compact if and only if every infinite set has an accumulation point .

For metrizable topological spaces the terms compactness, sequence compactness and countable compactness agree.

Looking at Hausdorff spaces , the countable compact spaces are clearly characterized by the theorem of Mazurkiewicz - Sierpinski : Every countable compact space is homeomorphic to a well-ordered set with an order topology .

The product of two countably compact spaces is in general not always countably compact. This is in clear contrast to Tychonoff's theorem , which says that the product of (even uncountably many) compact spaces is compact again.

example

From the above properties it immediately follows that every topology on a countable set makes it a countably compact space, for example natural numbers with a discrete topology are countably compact. If one considers an uncountable set with a co-countable topology , this topological space is uncountably compact.

If one considers the ordinal number space with order topology , where the first uncountable ordinal number denotes, then this topological space is countably compact. ${\displaystyle [0,\omega _{1})}$ ${\displaystyle \omega _{1}}$

literature

• René Bartsch: General Topology . 2nd Edition. De Gruyter , Berlin 2015, ISBN 978-3-11-040618-4 .
• John L. Kelley: General Topology . Springer-Verlag, New York 1955, ISBN 0-387-90125-6 .
• Philippe Blanchard, Erwin Brüning: Direct methods of the calculus of variations . Springer-Verlag, Vienna 1982, ISBN 0-387-81692-5 .

Individual evidence

1. ↑ Bartsch: General Topology. 2015, p. 142.
2. ↑ Bartsch: General Topology. 2015, p. 144.
3. ↑ Kelley: General Topology. 1955, p. 162.
4. ↑ Bartsch: General Topology. 2015, p. 147.
5. ^ Mazurkiewicz, Stefan, and Sierpiński, Wacław: Contribution à la topologie des ensembles dénombrables . In: Fundamenta Mathematicae . tape 1 , no. 1 , 1920, p. 17-27 ( eudml.org ).
6. ↑ Engelking: General Topology. 1989, Example 3.10.19.