Infinity valued partial metrics
Submitted: 2023-12-06
|Accepted: 2024-09-18
|Published: 2025-04-01
Copyright (c) 2024 Homeira Pajoohesh, Steve Matthews
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Downloads
Keywords:
partial metrics, infinite distance functions
Supporting agencies:
Abstract:
A partial metric is a generalised metric incorporating non-zero self-distance. It was introduced by Matthews in Computer Science, and connected well with Kopperman's long standing interest in generalised metrics and non-Hausdorff topology. In this article we further generalise partial metrics by incoporating ∞ as a distance. We prove that a so-called infinite partial metric induces a partial metrizable topology, but not in general vice verse.
References:
S. Assaf and K.Pal, Partial metric spaces with negative distances and fixed point theorems, Topology Proc. 49 (2017), 19-40.
G.Beer, The structure of infinite real-valued metric spaces, Set-Valued Var. Anal. 21, no.4 (2013), 591-602.
https://doi.org/10.1007/s11228-013-0255-2
M. Bukatin, R. Kopperman, S. Matthews and H.Pajoohesh, Partial metric spaces, American Mathematical Monthly 116 (2009), 708-718.
https://doi.org/10.4169/193009709X460831
M. Darnel and W. C. Holland, Minimal non-metabelian varieties of l-groups that contain no nonabelian o-groups, Communications in Algebra 42 (2014), 5100-5133.
https://doi.org/10.1080/00927872.2013.833209
M. Darnel, W. C.Holland, and H. Pajoohesh, The relationship of partial metric varieties and commuting powers varieties, Order 30, no.2 (2013), 403-414.
https://doi.org/10.1007/s11083-012-9251-7
C. Holland, R. Kopperman, and H. Pajoohesh, Intrinsic generalized metrics, Algebra Universalis 67, no. 1 (2012), 1-18.
https://doi.org/10.1007/s00012-012-0168-1
R. Kopperman, S. Matthews, and H. Pajoohesh, Universal partial metrizability, Applied General Topology 5 (2004), 115-127.
https://doi.org/10.4995/agt.2004.2000
R. Kopperman, All topologies come from generalised metrics, American Mathematical Monthly 95, no. 2 (1988).
https://doi.org/10.2307/2323060
S. Matthews, Partial metric topology, Proc. 8th summer conference on topology and its applications, ed S. Andima et al., New York Academy of Sciences Annals, 728,183-197, 1994.
https://doi.org/10.1111/j.1749-6632.1994.tb44144.x
S. Oltra and O. Valero, Banach's fixed point theorem for partial metric spaces, Rend. Istit. Mat. Univ. Trieste 36, no. 1-2 (2004), 17-26.
S. J. O'Neill, Partial metrics, valuations and domain theory, Proc. 11th Summer Conference on General Topology and Applications. Ann. New York Acad. Sci. 806, 304-315, 1996.
https://doi.org/10.1111/j.1749-6632.1996.tb49177.x
H.Pajoohesh, The relationship of partial metric varieties and commuting powers varieties II, Algebra Universalis, 73, no. 3-4 (2015), 291-295.
https://doi.org/10.1007/s00012-015-0331-6
H.Pajoohesh, $T_0$ functional Alexandroff topologies are partial metrizable, Applied General Topology 25, no. 2 (2024), 305-319.
https://doi.org/10.4995/agt.2024.19401
D. Scott, Outline of a mathematical theory of computation, Oxford University Computing Laboratory, PRG02, 1970.
J. E. Stoy, The Scott-Strachey Approach to Programming Language Theory, MIT Press 1977.
O. Valero, On Banach fixed point theorems for partial metric spaces, Applied General Topology 6, no. 2 (2005), 229-240.
