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A note on separation in AP

“…As an answer to a question raised by H. Herrlich, an internal description of the epireflector from TOP onto TOP 2 was given by V. Kannan in [5], making use of a transfinite construction. We will now derive an explicit description for the epireflector from AP onto AP 2 , along the same lines as was done for the T 1 -case in the section above.…”

“…In [11], three different suggestions for regularity were proposed, and in [5], it was motivated that the strongest one of them is the correct notion of regularity in the construct AP. We simply recall this definition for the sake of completeness.…”

Regularity for relational algebras and approach spaces

“…It was further studied by B. Banaschewski, R. Lowen and C. Van Olmen [1] and has interesting applications in the setting of extensions of contractions and function spaces, see Jäger [12] and Colebunders, Mynard, Troth [9]. A characterization of regularity generalizing (1.1) to the setting of approach spaces, which we recall in (5.1), was presented by Brock and Kent [4]. It is based on the limit operator acting on filters and on some diagonal operation on selected filters.…”

“…An approach space is called regular if it satisfies one and hence both conditions. Brock and Kent [4] have shown that regularity of an approach space is equivalent to the following conditions in terms of selections, one is a version based on filters, the other uses only ultrafilters. If A is a set, ψ : A → X, σ : A → F p X, and G ∈ F p A, then 5.2.…”