On Urysohn's Lemma for generalized topological spaces in ZF

A strong generalized topological space is an ordered pair $$\mathbf{X}$=\langle X, $\mathcal{T}$\rangle$suchthat$X$isasetand$$\mathcal{T}$$ is a collection of subsets of $X$ such that $\mathrm{\varnothing },X\in \mathcal{T}$ and $\mathcal{T}$ 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 $\mathbf{Z}\mathbf{F}$. Notions of a U-normal and an effectively normal generalized topological space are introduced. It is observed that, in $\mathbf{Z}\mathbf{F}+\mathbf{D}\mathbf{C}$, every U-normal generalized topological space satisfies Urysohn’s lemma. It is shown that every effectively normal generalized topological space satisfies Csasz'ar’s modification of Urysohn’s Lemma. A $\mathbf{Z}\mathbf{F}$- example of a strong generalized topological normal space which satisfies the Tietze-Urysohn Extension Theorem and fails to satisfy Urysohn’s Lemma is shown.