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. ${\ displaystyle (X, {\ mathcal {O}})}$ ${\ displaystyle {\ mathcal {S}}}$

Then are equivalent:

for every cover of with sets of there exists a finite partial cover ${\ displaystyle X}$ ${\ displaystyle {\ mathcal {O}}}$

for every cover of with sets of there exists a finite partial cover

{\ displaystyle X}

{\ displaystyle {\ mathcal {S}}}

In particular, it is sufficient to check compactness with the sets of the sub-base.

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 .