I am trying to write another proof (using my theory) of Urysohn lemma. This question has appeared during this research.

Let $\mu$ be a $T_4$ (normal) topology on some set $\mho$. Let $\delta$ be the proximity generated by this topology by the formula $X\delta Y\Leftrightarrow \overline X\cap\overline Y\ne\emptyset$ ($\overline X$ denotes topological closure of $X$).

By definition an entourage $U$ of $\delta$ is a binary relation on $\mho$ such that $X\delta Y\Rightarrow (X\times Y)\cap U\ne\emptyset$ for every sets $X,Y\in\mathscr{P}\mho$.

Conjecture If for some closed sets $A,B\in\mathscr{P}\mho$ we have $\lnot(A\delta B)$, then there exists an entourage $U$ of $\delta$ such that $(A\times B)\cap(U\circ U^{-1}) = \emptyset$.

$\circ$ denotes composition of binary relations: $g\circ f=\{ (x,z) \mid \exists y:((x,y)\in f\land (y,z)\in g \}$.


Let $V$ and $W$ be disjoint open nbhds of $A$ and $B$, respectively. Let

$$U=V^2\cup W^2\cup\big(\mho\setminus(A\cup B)\big)^2\;;$$

$U$ is an open nbhd of the diagonal in $\mho^2$, so it’s easily seen to be an entourage of $\delta$.

Suppose that $\langle a,b\rangle\in(A\times B)\cap(U\circ U^{-1})$; then there is a $z\in\mho$ such that $\langle a,z\rangle\in U^{-1}$ and $\langle z,b\rangle\in U$, i.e., such that $\langle z,a\rangle,\langle z,b\rangle\in U$.

If $z\in V$, then $z\notin W$, so $$\langle z,b\rangle\in V^2\cup\big(\mho\setminus(A\cup B)\big)^2\;,$$ and hence $b\in V\cup\big(\mho\setminus(A\cup B)\big)$, which is impossible. Similarly, $z\in W$ implies that $a\in W\cup\big(\mho\setminus(A\cup B)\big)$, and $z\in\mho\setminus(A\cup B)$ implies that $a,b\in\mho\setminus(A\cup B)$, both of which are also impossible. Thus, $(A\times B)\cap(U\circ U^{-1})=\varnothing$.


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