Convexities in T0-quasi-metric spaces

In this thesis, we recall three types of convexities in metric spaces, namely; Menger convexity, Takahashi convexity andM-convexity. We then generalise these convexities to the framework of T0-quasi-metric spaces. Since the concept of convexities heavily relies on the concept of betweenness, a fundamental concept in the study of axiomatic geometry, we begin by generalising the concept of betweenness to T0-quasi-metric spaces. We show that Takahashi convexity implies Menger convexity in T0-quasi-metric spaces. Lastly, we generalise the concept of M-convexity to T0-quasi-metric setting and present some best approximations in these spaces