A survey of the theory of M-spaces

✍ Scribed by Kiiti Morita

• Publisher: Elsevier Science
• Year: 1971
• Weight: 834 KB
• Volume: 1
• Category: Article
• ISSN: 0016-660X
• DOI: 10.1016/0016-660x(71)90110-3

✦ Synopsis

uced the notion of M-spaces in connection with the problem of cb cterizing a. space whose product. with any metric sparse is Iormal. S'nce then, man 'nteresling results have been obtained b ~r many matherr; aticians. e would hke to give a survey on these results.

It is assumed that spxes are T1-spaces, maps are continuous maps, and "patacompac Garaeompact

. We shall begin wit : the definition of M-spaces.

on [ 2 11. A qpace % is called an M-space i:f there is a narmal sequence ( l.Ii} 0 n coverings of X satisfying cofadition (M) below. creasing sequence of non-emp!;y closed :subsets of , Ui), then Ki # Q).

[2 1:) . A space X is an M-space (resp a paracompact A s a quasi-perfect (reqx perfect) map from X onto a metrx space. re a map f : X -+ Y is called a perfect (resp. quasi-perfect) map iff ;s closed, onto and if f-l (v) is compac (resp. cou 1:~ cournpactj foI each point J of Iy. Thus, metric spaces and colntably co an ara zom~~act sp xe X whi dh is 6, in