Type: Article

Continuous weak selections for products

García-Ferreira, S.; Miyazaki, K.; Nogura, T.;

Abstract:

A weak selection on an infinite set X is a function sigma : [X](2) -> X such that sigma({x, y}) is an element of {x, y} for each {x, y} is an element of [X](2). A weak selection on a space is said to be continuous if it is a continuous function with respect to the Vietoris topology on [X](2) and the topology on X. We study some topological consequences from the existence of a continuous weak selection on the product X x Y for the following particular cases: (i) Both X and Y are spaces with one non-isolated point. (ii) X is a space with one non-isolated point and Y is an ordinal space. As applications of the results obtained for these cases, we have that if X is the continuous closed image of suborderable space, Y is not discrete and has countable tightness, and X x Y admits a continuous weak selection, then X is hereditary paracompact. Also, if X is a space, Y is not-discrete and Sel(2)(c)(X x Y) not equal emptyset, then X is totally disconnected. (C) 2013 Elsevier B.V. All rights reserved.