On Urysohn's Lemma for generalized topological spaces in ZF

TL;DR

This paper investigates conditions under which Urysohn's lemma holds in generalized topological spaces within ZF set theory, introducing notions of normality and providing examples and variants of the lemma.

Contribution

It establishes necessary and sufficient conditions for Urysohn's lemma in generalized topological spaces in ZF and introduces new concepts like U-normal and effectively normal spaces.

Findings

Urysohn's lemma holds under certain conditions in ZF.

Every U-normal space satisfies Urysohn's lemma in ZF+DC.

An example of a space satisfying Tietze-Urysohn but not Urysohn's lemma is provided.

Abstract

A strong generalized topological space is an ordered pair $\mathbf{X}=\langle X, \mathcal{T}\rangle$ such that $X$ is a set and $\mathcal{T}$ is a collection of subsets of $X$ such that $\emptyset, X\in \mathcal{T}$ 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 $\mathbf{ZF}$. Notions of a U-normal and an effectively normal generalized topological space are introduced. It is observed that, in $\mathbf{ZF}+\mathbf{DC}$, 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 $\mathbf{ZF}$- example of a strong generalized topological normal space which satisfies the Tietze-Urysohn Extension Theorem and fails to satisfy Urysohn's Lemma is shown.