On open maps and related functions over the Salbany compactification

Nxumalo, Mbekezeli

Archivum mathematicum, Tome 60 (2024) no. 1, pp. 21-33.

Résumé

Given a topological space $X$, let $\mathcal{U}X$ and $\eta _{X}\colon X\rightarrow \mathcal{U}X$ denote, respectively, the Salbany compactification of $X$ and the compactification map called the Salbany map of $X$. For every continuous function $f\colon X\rightarrow Y$, there is a continuous function $\mathcal{U}f\colon \mathcal{U}X\rightarrow \mathcal{U}Y$, called the Salbany lift of $f$, satisfying $(\mathcal{U}f)\circ \eta _{X}=\eta _{Y}\circ f$. If a continuous function $f\colon X\rightarrow Y$ has a stably compact codomain $Y$, then there is a Salbany extension $F\colon \mathcal{U}X\rightarrow Y$ of $f$, not necessarily unique, such that $F\circ \eta _{X}=f$. In this paper, we give a condition on a space such that its Salbany map is open. In particular, we prove that in a class of Hausdorff spaces, the spaces with open Salbany maps are precisely those that are almost discrete. We also investigate openness of the Salbany lift and a Salbany extension of a continuous function. Related to open continuous functions are initial maps as well as nearly open maps. It turns out that the Salbany map of every space is both initial and nearly open. We repeat the procedure done for openness of Salbany maps, Salbany lifts and Salbany extensions to their initiality and nearly openness.

Détail

DOI : 10.5817/AM2024-1-21

Classification : 54D35, 54D80
Keywords: ultrafilter; ultrafilter space; compact space; compactification; open map; initial map; nearly open map; compact-open basis; spectral space; quasi-spectral space