RE-proximities as fixed points of an operator on pseudo proximities

Convergences, pretopologies and proximities induced by Cauchy and Riesz screens

“…As it was recalled in the proof of the preceding proposition, an ultrafilter U on X is in D(∆) if and only if γ(U) is in B [4]. Finally C rd is a screen if and only if for every x in X, γ(x) is a δ-round and compressed filter.…”

“…According to [4], under this condition, the proximity R(δ) associated to ∆ is an RE-proximity. It follows from the above discussion and Theorem 4.1.2 that this proximity is induced by the Cauchy screen C rd .…”

“…We have the following characterization from [4]: δ is an RE-proximity if and only if its nasse ∆ is the closure in (βX) 2 of its equivalence kernel ∆ if and only if R(δ) = δ. In particular if δ is an RE-proximity D(∆) is dense in βX and δ is induced by the Cauchy screen C rd .…”

“…Proposition 4.1.1[4]. For any proximity δ with associated nasse ∆, a filter F is δ-compressed if and only if β(F) × β(F) ⊂ ∆.Corollary 4.1.1.…”