Open AccessOn function spaces related to some kinds of weakly sober spaces
Author(s) -
Xiaoyuan Zhang,
Meng Bao,
Xiaoquan Xu
Publication year - 2022
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022516
Subject(s) - pointwise convergence , bounded function , topology (electrical circuits) , mathematics , pointwise , space (punctuation) , compact open topology , function space , function (biology) , weak topology (polar topology) , uniform continuity , topological space , pure mathematics , general topology , interpolation space , metric space , mathematical analysis , combinatorics , extension topology , network topology , functional analysis , computer science , biochemistry , chemistry , evolutionary biology , biology , gene , operating system
In this paper, we mainly study function spaces related to some kinds of weakly sober spaces, such as bounded sober spaces, $ k $-bounded sober spaces and weakly sober spaces. For $ T_{0} $ spaces $ X $ and $ Y $, it is proved that $ Y $ is bounded sober iff the function space $ {\bf{Top}}(X, Y) $ of all continuous functions $ f : X\longrightarrow Y $ equipped with the pointwise convergence topology is bounded sober iff $ {\bf{Top}}(X, Y) $ equipped with the Isbell topology is bounded sober. But for a $ k $-bounded sober space $ X $, the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology may not be $ k $-bounded sober. It is shown that if the function space $ {\bf{Top}}(X, Y) $ equipped with the pointwise convergence topology or the Isbell topology is weakly sober (resp., a cut space), then $ Y $ is weakly sober (resp., a cut space). Relationships among some kinds of (weakly) sober spaces are also investigated.