A study on quasi-pseudometrics

Natasha Demetriou, Hans-Peter A. Kunzi
2017 Hacettepe Journal of Mathematics and Statistics  
We study some aspects of the space QP M (X) of all quasipseudometrics on a set X equipped with the extended T0-quasi-metric AX (f, g) = sup (x,y)∈X×X (f (x, y)−g(x, y)) whenever f, g ∈ QP M (X). We observe that this space is bicomplete and exhibit various closed subspaces of (QP M (X), τ ((AX ) s )). In the second part of the paper, as a rough way to measure the asymmety of a quasi-pseudometric f on a set X, we investigate some properties of the value (AX ) s (f, f −1 ). The authors would like
more » ... authors would like to thank the South African National Research Foundation for partial nancial support under grants IFR1202200082 and CPRR14071175245. ‡ For a, b ∈ R we set a−b = max{a − b, 0} = (a − b) ∨ 0. The general construction of the inmum of two quasi-pseudometrics will be discussed briey below in the last section of this paper. † † Note that if d 1 , d 2 are quasi-pseudometrics, then s is a quasi-pseudometric, while b need not satisfy the triangle inequality, as Example 2 shows.
doi:10.15672/hjms.2016.396 fatcat:emct2s2qqbbdrbksdkaznamvji