Continuous weak selections for products

A weak selection on an infinite set X   is a function σ:[X]2→Xσ:[X]2→X such that σ({x,y})∈{x,y}σ({x,y})∈{x,y} for each {x,y}∈[X]2{x,y}∈[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[X]2 and the topology on X  . We study some topological consequences from the existence of a continuous weak selection on the product X×YX×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×YX×Y admits a continuous weak selection, then X is hereditary paracompact. Also, if X is a space, Y   is not-discrete and Sel2c(X×Y)≠∅, then X is totally disconnected.