Retraction in 𝑚-paracompact spaces

In this paper we show that such a method is available for the class of m-paracompact normal spaces, and more generally for any classes of normal spaces which can be characterized by the normality of their product with a compact Hausdorff space. We are basically motivated by Morita's characterization of an m-paracompact normal space (Theorem 1). In Lemma 1 we establish a sort of distributive property over products which the attaching of spaces possesses, and we are then easily able to prove that the adjunction space of two m-paracompact normal spaces is m-paracompact normal. Then by some known techniques, it follows that if Q is the class of m-paracompact normal spaces and XQ and X is an AR(Q) if and only if X is a contractible ANR(Q). 2. Preliminaries. Let m be an infinite cardinal number. A space X is m-paracompact if every open cover of cardinality <m has a locally finite open refinement. We let I denote the closed unit interval [0, 1], and Im denotes the product space of m copies of I. The reader is referred to [6] for some extensive results on m-paracompact spaces, and to [3] for definitions and basic properties of an AR(Q), resp. ANR(Q), i.e., absolute retract, resp. absolute neighborhood retract for a class Q of normal spaces and an ES(Q), resp. NES(Q), i.e., extension space, resp. neighborhood extension space for a class Q of normal spaces. Morita [6, p. 229] characterizes an m-paracompact normal space in the following useful way: