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 .
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.
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.
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
- ↑ Bartsch: General Topology. 2015, p. 142.
- ↑ Bartsch: General Topology. 2015, p. 144.
- ↑ Kelley: General Topology. 1955, p. 162.
- ↑ Bartsch: General Topology. 2015, p. 147.
- ^ 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 ).
- ↑ Engelking: General Topology. 1989, Example 3.10.19.