MIYAZAKI KAZUMI

We study properties of the pseudocompact spaces X with a weak selection, and we dedicate a particular attention to the weak selection topologies on X. In case when X is also locally compact, we obtain a convenient decomposition of X into a finite union of clopen sets, which are either almost compact or connected with a remainder of size two in their Stone-Cech compactification. (C) 2017 Elsevier B.V. All rights reserved.

A weak selection on an infinite set X is a function f : [X]2 → X such that f({x,y}) ∈ {x,y} for each {x,y} ∈ [X]2. A weak selection f on X defines a relation x f y if f({x,y}) = x whenever x,y ∈ X are distinct. The topology τf on X generated by the weak selection f is the one which has the family of all intervals (←x)f = {y ∈ X : y f x} and (x →)f = {y ∈ X : x f y} as a subbase. 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. The paper deals with topological spaces (X,τ) for which there is a set W of continuous weak selections satisfying τ = Vf∈W τf (we say that the topology of X is generated by continuous weak selections). We prove that for any infinite cardinal α, there exists a weakly orderable space whose topology cannot be generated by less than or equal to α-many continuous weak selections. We prove that any subspace of a space generated by continuous weak selections is also generated by continuous weak selections. Assuming that c is regular, we construct a suborderable space whose topology is generated by c-many continuous weak selections but not by less than c. Also, under the assumption of GCH, for every infinite successor cardinal α+ we construct a space X that is generated by α+-many continuous weak selections but cannot be generated by α-many selections. © 2013 University of Houston.

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 Sel2c(X×Y)≠θ, then X is totally disconnected. © 2013 Elsevier B.V.

relative propertyにおけるsemi-metricとrelative first countable 研究と結果

異なる３種類のrelative normal について、部分空間Yがどんな全空間の中においてもそれぞれのrelative normalになるときの特徴づけを示し、さらにrelative s-normalの場合には部分空間Yがcompactであるという同値条件を得た。これによりnormalの観点からのcompact 空間の利用範囲をさらに広げることができた。