Admissible topologies on $C(Y,Z)$ and ${\cal O}_Z(Y)$

TL;DR

This paper investigates Scott-type topologies on the lattice of open sets and constructs new admissible topologies on spaces of continuous functions and their inverse images, addressing open problems in topology.

Contribution

It introduces new admissible topologies on $C(Y,Z)$ and ${ m O}_Z(Y)$, expanding the understanding of topologies on function spaces.

Findings

Defined Scott-type topologies on ${ m O}(Y)$

Constructed admissible topologies on $C(Y,Z)$

Identified new problems in the topology of function spaces

Abstract

Let $Y$ and $Z$ be two given topological spaces, ${\cal O}(Y)$ (respectively, ${\cal O}(Z)$) the set of all open subsets of $Y$ (respectively, $Z$), and $C(Y,Z)$ the set of all continuous maps from $Y$ to $Z$. We study Scott type topologies on ${\mathcal O}(Y)$ and we construct admissible topologies on $C(Y,Z)$ and ${\mathcal O}_Z(Y)=\{f^{-1}(U)\in {\mathcal O}(Y): f\in C(Y,Z)\ {\rm and}\ U\in {\mathcal O}(Z)\}$, introducing new problems in the field.