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On a question of S. Godlewski

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Laguna, V.F. and Morón, Manuel A. and Rodríguez Sanjurjo, José Manuel (1990) On a question of S. Godlewski. Questions and answers in general topology , 8 (2). pp. 441-447. ISSN 0918-4732



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.


Item Type:Article
Uncontrolled Keywords:Mutational retract; MANR; mutational absolute neighborhood retract; space of components
Subjects:Sciences > Mathematics > Topology
ID Code:20479
Deposited On:19 Mar 2013 18:13
Last Modified:12 Dec 2018 15:13

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