Weak and strong structures and the $T_{3.5}$ property for generalized topological spaces

TL;DR

This paper explores the properties and structures of generalized topological spaces, introduces the $T_{3.5}$ class, and establishes their relationships with other topological concepts, including product theorems and continuity conditions.

Contribution

It introduces the $T_{3.5}$ generalized topological spaces, characterizes them via subspaces of powers of a natural topology on [0,1], and proves product theorems for normality, compactness, and Lindelöfness.

Findings

$T_{3.5}$ spaces are subspaces of powers of a natural topology on [0,1]

Normality, compactness, and Lindelöfness are productive in generalized topologies

Continuous maps between ordered generalized spaces are monotonic and order-continuous

Abstract

We investigate weak and strong structures for generalized topological spaces, among others products, sums, subspaces, quotients, and the complete lattice of generalized topologies on a given set. Also we introduce $T_{3.5}$ generalized topological spaces and give a necessary and sufficient condition for a generalized topological space to be a $T_{3.5}$ space: they are exactly the subspaces of powers of a certain natural generalized topology on $[0,1]$. For spaces with at least two points here we can have even dense subspaces. Also, $T_{3.5}$ generalized topological spaces are exactly the dense subspaces of compact $T_4$ generalized topological spaces. We show that normality is productive for generalized topological spaces. For compact generalized topological spaces we prove the analogue of the Tychonoff product theorem. We prove that also Lindel\"ofness (and $\kappa $-compactness) is productive for generalized topological spaces. On any ordered set we introduce a generalized topology and determine the continuous maps between two such generalized topological spaces: for $|X|, |Y| \ge 2$ they are the monotonous maps continuous between the respective order topologies. We investigate the relation of sums and subspaces of generalized topological spaces to ways of defining generalized topological spaces.