Morita theorem

The set of Morita is a theorem of mathematical sub-region of the topology . The sentence goes back to a scientific work by the Japanese mathematician Kiiti Morita from 1948 and deals with the problem under which conditions a topological space has the property of paracompactness . It is related to the theorem on metrisability and paracompactness by the British mathematician Arthur Harold Stone .

Formulation of the sentence

The sentence can be formulated as follows:

Under the general assumption of the countable axiom of choice :

Every regular Lindelöf room is paracompact.

The following applies in detail:

If a regular Lindelöf space and an arbitrary open covering of , then a sequence of open - subsets can be covered in such a way that a locally finite refinement of is formed. ${\displaystyle X}$ ${\displaystyle {\mathcal {U}}=\left(U_{i}\right)_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}=\left(O_{k}\right)_{k=1,2,3.\ldots }}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {O}}}$ ${\displaystyle {\mathcal {U}}}$

A somewhat different, but closely related formulation of the sentence can be found in the monograph Topology by James Dugundji . It says:

In a Hausdorff Lindelöf space, regularity and paracompactness are concepts of equal value.

Inferences

The following corollaries can be drawn from the Moritas theorem :

Corollary 1 ( Stone's theorem for separable spaces ) :

In a separable metric space , every open cover has a locally finite, countable refinement.

Corollary 2 :

A regular Hausdorff Lindelöf room is always a T ₄ room . This is especially true for every regular Hausdorff space that satisfies the second axiom of countability .

literature

• James Dugundji : Topology. 8th printing. Allyn and Bacon, Boston MA 1973.
• Ernest Michael : A note on paracompact spaces . In: Proceedings of the American Mathematical Society . tape 4 , no. 5 , 1953, pp. 831-838 , JSTOR : 2032419 .
• Kiiti Morita : Star-finite coverings and the star-finite property . In: Mathematica Japonica . tape 1 , 1948, p. 60-68 .
• 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 ).
• Martin Väth : Topological Analysis. From the Basics to the Triple Degree for Nonlinear Fredholm Inclusions (=  De Gruyter Series in Nonlinear Analysis and Applications . Volume 16 ). Walter de Gruyter, Berlin / Boston 2012, ISBN 978-3-11-027722-7 ( MR2961860 ).
• Stephen Willard : General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA et al. a. 1970, ISBN 0-201-08707-3 ( MR0264581 ).

Individual evidence

1. ^ Martin Väth: Topological Analysis. 2012, p. 96 ff.
2. ^ Martin Väth: Topological Analysis. 2012, p. 96.
3. ↑ Stephen Willard: General Topology. 1970, p. 146
4. ↑ James Dugundji: Topology. 1973, pp. 174-175
5. ^ Martin Väth: Topological Analysis. 2012, pp. 97-98.