On Urysohn’s Lemma for Generalized Topological Spaces in ${{\mathbf {ZF}}}$

Abstract

A strong generalized topological space is an ordered pair ${\mathbf {X}}=\langle X, \mu \rangle$ such that X is a set and $\mu$ is a collection of subsets of X which covers X and is stable under arbitrary unions. A necessary and sufficient condition for a strong generalized topological space ${\mathbf {X}}$ to satisfy Urysohn’s lemma or its appropriate variant is shown in $\mathbf {ZF}$. Notions of a U-normal and an effectively normal generalized topological space are introduced. It is observed that, in $\mathbf {ZF}$, the Principle of Dependent Choices implies that every U-normal generalized topological space satisfies Urysohn’s lemma. It is shown that every effectively normal generalized topological space satisfies Csaszár’s modification of Urysohn’s lemma. In $\mathbf {ZFA}$ (also in $\mathbf {ZF}$), it is proved that the Axiom of Choice is equivalent to the statement “Every normal strong generalized topological space is effectively normal”. A $\mathbf {ZF}$-example of a strong generalized topological normal space which satisfies the Tietze–Urysohn Extension Theorem and fails to satisfy Urysohn’s lemma is shown. Several intriguing open problems are posed.