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 . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle A\subseteq X}$ ${\displaystyle B\subseteq Y}$ ${\displaystyle W}$ ${\displaystyle A\times B}$ ${\displaystyle X\times Y}$

Then there are open subsets and with . ${\displaystyle U\subseteq X}$ ${\displaystyle V\subseteq Y}$ ${\displaystyle A\times B\subseteq U\times V\subseteq W}$

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. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle X}$ ${\displaystyle A\times B\subset W:=\{(x,y)\in X\times X;x\not =y\}\subset X\times X}$ ${\displaystyle X}$ ${\displaystyle W}$ ${\displaystyle U\supseteq A}$ ${\displaystyle V\supseteq B}$ ${\displaystyle U\times V\subseteq W}$ ${\displaystyle U\cap V=\emptyset }$ ${\displaystyle X}$

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 .

Individual evidence

1. ↑ For more on the vita, see here .
2. ↑ Kelley: General topology. 1975, p. 142.
3. ↑ Shershin: Introduction to topological semi groups. 1979, p. 23.
4. ^ Joshi: Introduction to General Topology. 1983, p. 171.
5. ↑ Shershin: Introduction to topological semi groups. 1979, p. 24.