RE-proximities as fixed points of an operator on pseudo proximities (Q1568398)

A topological space \(Y\) is called a Riesz extension of \(\delta\), a compatible proximity on \(X\), provided \(A\delta B\) in \(X\) if and only if \(Y\) contains a copy of \(X\) and the closures in \(Y\) of \(A\) and \(B\) intersect. That such an extension exists for an EF-proximity, with \(Y\) a Hausdorff compactification of \(X\), was shown by \textit{Yu. M. Smirnov} [Am. Math. Soc., Transl., II. Ser. 38, 5-35 (1964); translation from Mat. Sb., N. Ser. 31(73), 543-574 (1952; Zbl 0047.41903)]. This result was generalized to LO-spaces by the combined efforts of \textit{M. S. Gagrat} and \textit{S. A. Naimpally} [Fundam. Math. 71, 63-76 (1971; Zbl 0188.27902)] and \textit{W. J. Thron} [Proc. Am. Math. Soc. 40, 323-326 (1973; Zbl 0249.54014)]. \textit{Á. Császár} [Acta Math. Hung. 47, 201-221 (1986; Zbl 0611.54019)] studied RE-proximity \(\delta\) wherein \(Y\) is regular. Let \(\Omega(X)\) denote the set of all ultrafilters on \(X\). Then \(\Delta= \{(\mathbb{U},\mathbb{V})\in \Omega(X) \times\Omega (X):\mathbb{U} \times\mathbb{V}\subset \delta\}\) is called the nasse of \(\delta\). \textit{L. Haddad} [Ann. Fac. Sci. Univ. Clermont 44, Math. 7, 3-80 (1970; Zbl 0224.54002)] showed that a pseudoproximity \(\delta\) is EF if and only if the nasse \(\Delta\) is an equivalence relation on \(\Omega(X)\). In this paper, with a natural idempotent operator, the author proves a similar result for RE-proximities.