| Popis: |
A weak selection on an infinite set X is a function σ : [ X ] 2 → X such that σ ( { x , y } ) ∈ { x , y } for each { 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 and the topology on X. We study some topological consequences from the existence of a continuous weak selection on the product 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 × 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 × Y ) ≠ ∅ , then X is totally disconnected. |
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