Alexander's theorem (set theoretical topology)
The set of Alexander is a mathematical theorem in the set topology . It provides a simplified criterion for checking the existence of finite partial covers with open sets in topological spaces and thus simplifies the proof of compactness .
The theorem was shown by James Waddell to Alexander II and is also known as Alexander subbasis lemma (Alexander's sub base lemma).
statement
Given a topological space and a sub-basis of the topology.
Then are equivalent:
- for every cover of with sets of there exists a finite partial cover
- for every cover of with sets of there exists a finite partial cover
In particular, it is sufficient to check compactness with the sets of the sub-base.
literature
- Steven Roman: Lattices and Ordered Sets . Springer, 2008, ISBN 978-0-387-78900-2 , doi : 10.1007 / 978-0-387-78901-9 .
- Boto von Querenburg : Set theoretical topology . 3. Edition. Springer-Verlag, Berlin Heidelberg New York 2001, ISBN 978-3-540-67790-1 , p. 106-107, Theorem 8.4 , doi : 10.1007 / 978-3-642-56860-2 .
Individual evidence
- ^ Roman: Lattices and Ordered Sets. 2008, p. 279.