Wallace's theorem
The set of Wallace is a theorem from the mathematical branch of topology , which the American mathematician Alexander Doniphan Wallace returns (1905-1985). It deals with a special separation property of compact product subspaces in product topologies : A product of compact sets in an open set lies in a product of open sets contained therein.
Formulation of the sentence
Given are two topological spaces and and embedded therein two compact subspaces and . Furthermore, let be an open superset of in .
Then there are open subsets and with .
Corollary
Every compact Hausdorff room is normal .
Namely, if and are closed, disjoint subsets of the compact Hausdorff space , then is . Since it is a Hausdorff room, the diagonal is closed, so it is open. Applying Wallace's theorem above, we get two open sets and with , i. H. . This is normal.
literature
- John L. Kelley : General topology (= Graduate Texts in Mathematics . Volume 27 ). Reprint of the 1955 edition published by Van Nostrand. Springer, New York NY a. a. 1975, ISBN 3-540-90125-6 .
- Anthony Connors Shershin: Introduction to topological semigroups . University Presses of Florida, Miami FL 1979, ISBN 0-8130-0664-3 .
- Kapil D. Joshi: Introduction to General Topology . Wiley Eastern, New Delhi et al. a. 1983, ISBN 0-85226-444-5 .