Tamano's theorem

The set of Tamano is a theorem from the mathematical sub-region of the topology , the on Japanese mathematician Hisahiro Tamano back. He characterizes the paracompactness of topological spaces using the concepts of normality and compactness , including the Stone-Čech compactification .

Formulation of the sentence

The following conditions are equivalent for every Hausdorff room : ${\displaystyle X}$

1. ${\displaystyle X}$is paracompact .
2. ${\displaystyle X}$is completely regular and the topological product of with its Stone-Čech compactification is normal . ${\displaystyle X\times {\beta {X}}}$ ${\displaystyle X}$ ${\displaystyle \beta {X}}$
3. The topological product of with any compact Hausdorff space is normal . ${\displaystyle X\times K}$ ${\displaystyle X}$ ${\displaystyle K}$

Corollary

For every paracompact Hausdorff space and every compact Hausdorff space , the topological product is a paracompact Hausdorff space . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\times Y}$

This follows immediately with (3) and Tychonoff's theorem . This corollary in turn has the following result:

The following two conditions are equivalent for every Hausdorff room : ${\displaystyle X}$

1. ${\displaystyle X\times Y}$is normal for every paracompact Hausdorff room . ${\displaystyle Y}$
2. ${\displaystyle X\times Y}$is paracompact for every paracompact Hausdorff room . ${\displaystyle Y}$

literature

items

• Hisahiro Tamano: On Paracompactness . In: Pacific Journal of Mathematics . tape 10 , no. 3 , 1960, p. 1043-1047 , doi : 10.2140 / pjm.1960.10.1043 .

Monographs

• Gregory Naber: Set-theoretic Topology. With emphasis on problems from the theory of coverings, zero dimensionality and cardinal invariants . University Microfilms International, Ann Arbor MI 1977, ISBN 0-8357-0257-X .
• Jun-iti Nagata : Modern General Topology (=  North Holland Mathematical Library . Volume 33 ). 2nd revised edition. North-Holland Publishing, Amsterdam / New York / Oxford 1985, ISBN 0-444-87655-3 ( MR0831659 ).
• Horst Schubert : Topology. An introduction . 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 .
• Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. a. 1970.

Individual evidence

1. ↑ Tamano: On Para Compactness. 1960, pp. 1043-1047.
2. ^ Naber: Set-theoretic Topology. 1977, p. 161.
3. ↑ Nagata: Modern General Topology. 1985, p. 237.
4. ^ Willard: General Topology. 1970, p. 154.
5. ^ Naber: Set-theoretic Topology. 1977, p. 148.
6. ↑ Nagata: Modern General Topology. 1985, p. 223.
7. ^ Schubert: Topology. 1975, p. 85.
8. ^ Naber: Set-theoretic Topology. 1977, p. 163.