Isbell convexity in fuzzy quasi-metric spaces.

The concept of hyperconvexity in metric spaces was introduced by Aronszajn and Panichpakdi
in 1956. This concept was then generalised to the framework of quasi-metric spaces
by John Isbell in 1964, which he called Isbell convexity. In 2019, Yi git and Efe generalised
this concept of hyperconvexity to the framework of fuzzy metric spaces and they called this
new concept fuzzy hyperconvexity. In this MSc thesis, we introduce the concept of Isbell
convexity in fuzzy quasi-metric spaces, which we call fuzzy Isbell convexity. This idea extends
Isbell convexity in quasi-metric spaces to fuzzy quasi-metric spaces. We prove that
a fuzzy quasi-metric space is fuzzy Isbell convex if and only if it is fuzzy metrically convex
and has a mixed binary intersection property. Furthermore, we present the concept of a
compatible quasi-metric, which generalises the concept of the compatible metric introduced
by Radu, to the assymetric setting. We then use this new concept to generalise some xed
point theorems in quasi-metric spaces to the framework of fuzzy quasi-metric spaces. Finally,
we introduce a t-nonexpansive map and show that the xed point set of a t-nonexpansive
map in an F-bounded fuzzy Isbell convex space is fuzzy Isbell convex.