Cauchy structures and (\(m\)-)contiguities (Q1271922)

The author studies the interesting role of proximities to \((m)\)-contiguites. Firstly, considering CR- and CL-screens, i.e. Cauchy screens that are Riesz or Lobado, respectively, he generalizes some of his previous results. In the second section, the author proves the following interesting theorem: If a closure \(c\) is an \(S_1\)-closure that is Fréchet, the \({\mathcal R}^\circ(c)\) admits a compatible Cauchy screen provided \(| x| \leq 2^{2^\omega}\). The main theorem in Section 3 (after a series of useful lemmas) establishes the fundamental role of uniform \(m\)- (\(\omega\)-)loosenesses on the Cauchy property of \({\mathcal G}^\circ (m)\) or \({\mathcal G}^\circ({\mathcal R})\). Also, by the same theorem, one can see that the Efremovich and the strong Efremovich properties coincide for \(n\)-loosenesses. In the last section, the author deals with the problem of finding necessary and/or sufficient conditions for the screen \(\mathcal G(\mathcal T)\), where \(\mathcal T\) is a given looseness, composed of all \(\mathcal T\)-compressed filters to be Cauchy. He proves that uniform or (strongly) Efremovich loosenesses produce some sufficient conditions while none of them is necessary and he establishes that by offering some nice examples.