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Properties of domain representations of spaces through dyadic subbases

Published online by Cambridge University Press:  23 June 2016

YASUYUKI TSUKAMOTO and
HIDEKI TSUIKI
YASUYUKI TSUKAMOTO
Affiliation:
Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan Email: tsukamoto@i.h.kyoto-u.ac.jp and tsuiki@i.h.kyoto-u.ac.jp
HIDEKI TSUIKI
Affiliation:
Graduate School of Human and Environmental Studies, Kyoto University, Yoshida-nihonmatsu-cho, Sakyo-ku, Kyoto 606-8501, Japan Email: tsukamoto@i.h.kyoto-u.ac.jp and tsuiki@i.h.kyoto-u.ac.jp
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Abstract

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A dyadic subbase S of a topological space X is a subbase consisting of a countable collection of pairs of open subsets that are exteriors of each other. If a dyadic subbase S is proper, then we can construct a dcpo DS in which X is embedded. We study properties of S with respect to two aspects. (i) Whether the dcpo DS is consistently complete depends on not only S itself but also the enumeration of S. We give a characterization of S that induces the consistent completeness of DS regardless of its enumeration. (ii) If the space X is regular Hausdorff, then X is embedded in the minimal limit set of DS . We construct an example of a Hausdorff but non-regular space with a dyadic subbase S such that the minimal limit set of DS is empty.

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