On Function Spaces Related to H-sober Spaces

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 $T_{0}$ spaces $X$ and $Y$, it is proved that $Y$ is H-sober iff the function space $\mathbb{C}(X, Y)$ of all continuous functions $f : X\longrightarrow Y$ equipped with the topology of pointwise convergence is H-sober iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is H-sober. One immediate corollary is that for a $T_{0}$ space $X$, $Y$ is a sober space (resp., $d$-space, well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the topology of pointwise convergence is a sober space (resp., $d$-space, well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the the Isbell topology is a sober space (resp., $d$-space, well-filtered space). It is shown that $T_{0}$ spaces $X$ and $Y$, if the function space $\mathbb{C}(X, Y)$ equipped with the compact-open topology is H-sober, then $Y$ is H-sober. The function space $\mathbb{C}(X, Y)$ equipped with the Scott topology is also discussed.