On Function Spaces Related to H-sober Spaces
Meng Bao, Xiaoyuan Zhang, Xiaoquan Xu

TL;DR
This paper investigates the conditions under which various function space topologies preserve H-sobriety and related properties in T0 spaces, establishing equivalences and implications for pointwise, Isbell, compact-open, and Scott topologies.
Contribution
It provides new characterizations of H-sobriety preservation in function spaces with different topologies, linking properties of the codomain space to those of the function space.
Findings
Y is H-sober iff C(X,Y) with pointwise convergence is H-sober
Y is H-sober iff C(X,Y) with Isbell topology is H-sober
If C(X,Y) with compact-open topology is H-sober, then Y is H-sober
Abstract
In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and spaces and , it is proved that is H-sober iff the function space of all continuous functions equipped with the topology of pointwise convergence is H-sober iff the function space equipped with the Isbell topology is H-sober. One immediate corollary is that for a space , is a sober space (resp., -space, well-filtered space) iff the function space equipped with the topology of pointwise convergence is a sober space (resp., -space, well-filtered space) iff the function space equipped with the the Isbell topology is a sober space (resp., -space, well-filtered space). It is shown that spaces and , if the function space…
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Taxonomy
TopicsFuzzy and Soft Set Theory · Advanced Algebra and Logic · Rings, Modules, and Algebras
