We introduce the concept of a quasi-pseudometric type space and prove some fixed point theorems. Moreover, we connect this concept to the existing notion of quasi-cone metric space.
The concept of the tight extension of a metric space was introduced and studied by Dress. It is known that Dress theory is equivalent to the theory of the injective hull of a metric space independently discussed by Isbell some years earlier. Dress showed in particular that for a metric space the tight extension is maximal among the tight extensions of . In a previous work with P. Haihambo and H.-P. Künzi, we constructed the tight extension of a 0 -quasi-metric space. In this paper, we continue these investigations by presenting a similar construction in the category of -metric spaces and nonexpansive maps.
We continue our work on endpoints and startpoints inT0-quasimetric spaces. In particular we specialize some of our earlier results to the case of two-valuedT0-quasimetrics, that is, essentially, to partial orders. For instance, we observe that in a complete lattice the startpoints (resp., endpoints) in our sense are exactly the completely join-irreducible (resp., completely meet-irreducible) elements. We also discuss for a partially ordered set the connection between its Dedekind-MacNeille completion and theq-hyperconvex hull of its naturalT0-quasimetric space.
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