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 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。

In their study of relative metric spaces, Arkhangel'skii and Gordienko introduce several relative properties of paracompactness type. We investigate the relationships between these properties and other relative properties of paracompactness type and answer two questions raised by Arkhangel'skii and Gordienko. Theorem. If Y is strongly regular in X and 3 paracompact in X then Y is star normal in X. Theorem. Y is strongly star normal in X if and only if Y is Aull paracompact in X. Corollary. If Y is strongly star normal in X then Y is 2-paracompact in X. Corollary. If Y is properly metrizable in X then Y is strongly star normal in X. (c) 2005 Published by Elsevier B.V.

これまで研究してきたrelative paracompact, relative metacompact, relative normal, rerative σ-paracompactの関係を集約、詳しく図式化しそれらの間にあるgapとしての多くの反例を示した。さらにこの時点で解明できていない問題を公表した。

paracompact→collectionwise normal、metacompact + collectionwise normal→paracompactという２つのoriginal versionにおける関係を様々なrelative versionにおいて研究し、あるrelative collectionwise normalにおいては、はじめの図式が成り立たないという反例を示した。これによりoriginalでの常識が成り立つ境界を明らかにした。

relative paracompact,relative metacompactとoriginal versionのparacompact, matacompactを対比しこれを用いて他の性質への特徴づけを示した。さらに closurepreservingという観点から新しいrelative paracompactとrelative metacompact を定義し他のrelative propertyの中での位置と役割を示した。

We prove that a countable regular space has a continuous selection if and only if it is scattered. Further we prove that a paracompact scattered space admits a continuous selection if each of its points has a countable pseudo-base. We also provide two examples to show that: (1) paracompactness can not be replaced by countable compactness even together with (collectionwise) normality, and (2) having countable pseudo-base at each of its points can not be omitted even in the class of regular Lindelof linearly ordered spaces.

「compact空間においてhyperspaceへのcontinuous selectionをもつ同値条件はorderableである」というvan MillとWattelのよく知られている結果の“compact”を“almost compact”に弱めても同じ結果が得られることを示し、他の位相を導入したhyperspaceにも同様の結果が得られることを示した。

We give a characterization of countably paracompact and collectionwise normal spaces by means of set-valued semi-continuous selections. This provides a positive answer to a problem of V. Gutev.

Michaelのparacompact spaceにおける可算な局所有限開細分を用いた特徴づけがrelative paracompactにおいてどこまで成り立つかを調べ、成り立たない場合の反例も示した。更にcompact, Lindelof, paracompact spaceをrelative paracompactを用いて特徴づけした。