On a question of S. Godlewski

Abstract

S. Godlewski [Fundam. Math. 114, 1-9 (1981; Zbl 0498.54016)] proved that if a metrizable space X is a mutational retract of X0 and X is an MANR (mutational absolute neighborhood retract), then every component of X is a mutational retract of a component of X0. In the same paper he posed the problem whether this result remains true without the hypothesis X 2 MANR. In the present paper a partial answer is given to this question. A topological space is said to belong to the class U0 if a) the quotient map X ! (X) is closed (where (X) denotes the space of components) and b) ind((X)) = 0, where ind represents the small inductive dimension. Examples of classes of spaces contained in U0 include the MANR’s and the class of the locally compact metrizable spaces with compact components. The main results in this paper are: Theorem 6. Let X0 be a metrizable space and suppose that X is a mutational retract of X0. If X 2 U0 then every component of X is a mutational retract of a component of X0. - Proposition 8. Let X0 2 U0 be a metrizable space and let us suppose that X is a mutational retract of X0. Then X 2 U0. From these two results the following consequence is derived. Corollary 9. Let X0 2 U0 be a metrizable space and suppose that X is a mutational retract of X0. Then every component of X is a mutational retract of a component of X0. This corollary solves Godlewski’s problem provided the ambient space X0 belongs to the class U0.