On hereditary precompactness and completeness in quasi-uniform spaces

Abstract

Let X and Y be topological spaces, F (X, Y ) the set of all functions from X into Y and C(X, Y ) the set of all continuous functions in F (X, Y ). We study various set-open topologies t λ (λ ⊆ P(X)) on F (X, Y ) and consider their existence, comparison and coincidence in the setting of Y a general topological space as well as for Y = R. Further, we consider the parallel notion of quasi-uniform convergence topologies U λ (λ ⊆ P(X)) on F (X, Y ) to discuss U λ -closedness and right U λ -K-completeness properties of a certain subspace of F (X, Y ) in the case of Y a locally symmetric quasi-uniform space. We include some counter-examples to justify our comments.2010 MSC: 54C35; 46A16; 54E15; 54C08.

Quasi-uniform convergence topologies on function spaces- Revisited

“…In particular, a net {y α : α ∈ D} in a quasi-uniform or locally uniform space (Y, U) is said to be T (U)-convergent to y ∈ Y if, for each U ∈ U, there exists an α 0 ∈ D such that y α ∈ U [y] for all α ≥ α 0 . Definition 2.6 ( [41,26,24,7]). Let (Y, U) be a quasi-uniform space.…”

Quasi-uniform isomorphisms in fuzzy quasi-metric spaces, bicompletion and D-completion

“…In this direction, we recall that the notion of a quiet quasiuniform space introduced by Doitchinov in [8] provides a consistent theory of quasi-uniform completion which is different from the theory of bicompletion but it is very interesting and useful in many aspects (see [2,3,5,8], etc.). Furthermore, quietness provides a suitable framework to discuss many interesting properties in the realm of quasi-uniform spaces (see, for instance, [5,6,23,25,28]). In the light of these facts one can expect that an appropriate notion of quietness in the fuzzy setting provides a useful tool in the theory of fuzzy quasi-metric spaces.…”

On hyperspace topologies via distance functionals in quasi-metric spaces

“…Recall that a filter F on a quasi-uniform space (X, U) is a right K-Cauchy (resp. left K-Cauchy) [25], if for every U ∈ U there is an F ∈ F such that U −1 (x) ∈ F (resp. U (x) ∈ F) for all x ∈ F .…”