Scattered Spaces and the Class Mobi

H. R. Bennett, J. Chaber
1989 Proceedings of the American Mathematical Society  
We show that every regular scattered space with a point-countable base is an open and compact image of a scattered metacompact Moore space and, consequently, is an element of the minimal class of regular spaces containing all the scattered metric spaces and invariant under open and compact mappings. This gives a characterization of a subclass of the class MOBI. Let MOBI denote the minimal class of regular spaces containing all the metric spaces and invariant under open and compact mappings (see
more » ... [A]). It is easy to observe that a regular space is in MOBI if and only if it can be obtained as an image of a metric space under a mapping which is a composition of a finite number of open and compact mappings with regular domains (see [B]). The first exaxmples of nonmetacompact spaes in MOBI were constructed in [Ch], These examples have recently been improved by a construction in [BCh] of nonweakly 0-refinable space in MOBI. A common feature of these examples is that the patheological spaces in MOBI are scattered. Let MOBI (scattered) denote the class of all the regular spaces which can be obtained as images of scattered metric spaces under mappings which are compositions of a finite number of open and compact mappings with regular domains. The aim of this note is to prove Theorem. For a regular space Y the following conditions are equivalent. (a) Y is a MOBI (scattered), (b) Y is a scattered space in MOBI, (c) Y is a scattered space with a point-countable base.
doi:10.2307/2047394 fatcat:4kib72bgpjbelaih23hidc6upm