Linearly ordered collections and paracompactness

Abstract

Let us now define all of the terms which are mentioned in Theorem 1. Let cUt and 'U be collections of subsets of a topological space. A collection cUt endowed with a linear ( = total) order is said to be locally finite with respect to < provided that every majorized subcollection (that is, every subcollection of cU having an upper bound with respect to <) is locally finite. This definition is equivalent to that used by H. Tamano in [7] where he proved that (a) and (b) in Theorem 1 are equivalent in completely regular spaces. A collection 'u endowed with a linear order is said to be closure-preserving with respect to < provided that every majorized subcollection of CUt is closure-preserving. In order to define the term mentioned in (d), we first restate a definition given in [5]. A collection cit is said to be in a collection eu with cushion map f: cU-)V provided for every subcollection cUt' of cU we have cl(UcU') CUf('t'). We say that a collection 'U endowed with a linear order is in a collection 'V with cushion map f: cU--V provided for every majorized subcollection 9U' of U. we have cl(U'U') CUf(ult). We will omit explicit mention of the cushion map if no confusion will result. The above definitions of closure-preserving and linearly cushioned differ from those given by H. Tamano in [8] where he