A quasitopos containing CONV and MET as full subcategories

Abstract

ABSTRACT. We show that convergence spaces with continuous maps and metric spaces with contractions, can be viewed as entities of the same kind. Both can be characterized by a "limit function" % which with each filter associates a map % from the underlying set to {ne extended positive real line. Continuous maps and contractions can both be (htacterized as limit function preserving maps.lhe properties common to both the convergence and metric case serve as a basis for tne definition of the category, CAP. We show that CAP is a quasitopos and that, apart from the categories CONV, of convergence spaces, and MET, of metric spaces, it also contains the category AP of approach spaces as nicely embedded subcategorles.

Probabilistic convergence spaces and generalized metric spaces

“…The latter spaces were first introduced by G. Richardson and D. Kent [14] as generalizations of probabilistic metric spaces (see [15], [16]). The existence of an isomorphism between pqs-MET and a subcategory of PPRS follows from results established in [3] and [10]. Among other important full subcategories of PPRS are the categories PRTOP of pretopological spaces and TOP of topological spaces, each of which is embedded in PPRS in an obvious way.…”

A note on separation in AP

“…Constructing the T 1 −epireflector will carry us outside of AP, into the superconstruct PRAP of pre-approach spaces and contractions, as introduced in [9]. Let us only recall that a pre-approach space is a pair (X, δ) with δ : X × 2 X −→ [0, ∞] satisfying (D1), (D2) and (D3) (such δ is called a pre-distance) and contractions are defined in the same way as above.…”

“…a pre-approach system A, a pre-hull h and a pre-approach limit λ. For details we refer to [9], but we note that, just as for distances, stepping from AP to PRAP comes down to dropping the triangular axiom for A and λ and the idempotency for h. It was proved in [9] that AP is a concretely bireflective subconstruct of PRAP. Take (X, h) ∈ |PRAP|.…”

Local compactness in approach spaces I

“…As for topological spaces a convergence theory can be developed in AP (see E. and R. Lowen [5,6] for more details). The difference with topological spaces however is that with each filter and each point we can give a distance the point "is away from being a limit point" of the filter.…”