We study some aspects of the space $QPM(X)$ of all quasi-pseudometrics on a set $X$ equipped with the extended $T_0$-quasi-metric $A_X(f,g)=\sup_{(x,y)\in X\times X}(f(x,y)-g(x,y))$ whenever $f,g\in QPM(X)$. We observe that this space is bicomplete and exhibit various closed subspaces of $( QPM(X), \tau((A_X)^s))$.
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 $(A_X)^s(f,f^{-1}).
C.A. Agyingi, P. Haihambo and H.-P.A. Künzi, Tight extensions of T0-quasi-metric
spaces, in: V. Brattka, H. Diener, D. Spreen (Eds.), Logic, Computation, Hierarchies,
Festschrift in Honour of V.L. Selivanov's 60th Birthday, Ontos Verlag, De Gruyter Berlin,
Boston, 2014, pp. 922.
C.A. Agyingi, P. Haihambo and H.-P.A. Künzi, Endpoints in T0-quasi-metric spaces: Part
II, Abstract and Applied Analysis, Vol. 2013, Article ID 539573, 10 pages.
G. Berthiaume, On quasi-uniformities in hyperspaces, Proc. Amer. Math. Soc. 66 (1977),
335343.
M. Bukatin, R. Kopperman and S. Matthews, Some corollaries of the correspondence
between partial metrics and multivalued equalities, Fuzzy Sets Systems 256 (2014), 57
72.
M.J. Campión, E. Induráin, G. Ochoa and O. Valero, Functional equations related to
weighable quasi-metrics, Hacettepe J. Mat. Stat. 44 (4) (2015), 775787.
Ş. Cobzaş, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics,
Springer, Basel, 2012.
P. Fletcher and W.F. Lindgren, Quasi-uniform Spaces, Dekker, New York, 1982.
Y.U. Gaba and H.-P.A. Künzi, Splitting metrics by T0-quasi-metrics, Topology Appl. 193
(2015), 8496.
Y.U. Gaba and H.-P.A. Künzi, Partially ordered metric spaces produced by T0-quasimetrics,
Topology Appl. 202 (2016), 366383.
J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory, Selected Topics in
Point-Set Topology, Cambridge University Press, Cambridge, 2013.
D. Gronau, A remark on Sincov's functional equation, Not. S. Afr. Math. Soc. 31 (2000),
1-8.
E. Kemajou, H.-P.A. Künzi and O.O. Otafudu, The Isbell-hull of a di-space, Topology
Appl. 159 (2012), 24632475.
H.-P.A. Künzi, An introduction to quasi-uniform spaces, Beyond Topology, Contemp.
Math. 486 (2009), 239304.
H.-P.A. Künzi and C. Makitu Kivuvu, A double completion for an arbitrary T0-quasimetric
space, J. Logic Algebraic Programming 76 (2008), 251269.
H.-P.A. Künzi and S. Romaguera, Weightable quasi-uniformities, Acta Math. Hungar.
136 (1-2), (2012), 107128.
H.-P.A. Künzi and C. Ryser, The Bourbaki quasi-uniformity, Topology Proceedings 20
(1995), 161183.
H.-P.A. Künzi and M. Sanchis, The Katetov construction modified for a T0-quasi-metric
space, Topology Appl. 159 (2012), 711720.
H.-P.A. Künzi and V. Vajner, Weighted quasi-metrics, Ann. New York Acad. Sci. 728
(1994), 6477.
H.-P. A. Künzi and F. Yıldız, Convexity structures in T0-quasi-metric spaces, Topology
Appl. 200 (2016), 218.
L. Nel, Continuity Theory, Springer, Switzerland, 2016.
F. Plastria, Asymmetric distances, semidirected networks and majority in Fermat-Weber
problems, Ann. Oper. Res. (2009) 167: 121155.
I.V. Protasov, Coronas of balleans, Topology Appl. 149 (2005), 149160.
T. Salát, J. Tóth and L. Zsilinszky, Metric space of metrics defined on a given set, Real
Anal Exch., 18 no. 1 (19921993), 225231.
T. Salát, J. Tóth and L. Zsilinszky, On the structure of the space of metrics defined on a
given set, Real Anal. Exch. 19, no. 1 (19931994), 321327.
Year 2017,
Volume: 46 Issue: 1, 33 - 52, 01.02.2017
C.A. Agyingi, P. Haihambo and H.-P.A. Künzi, Tight extensions of T0-quasi-metric
spaces, in: V. Brattka, H. Diener, D. Spreen (Eds.), Logic, Computation, Hierarchies,
Festschrift in Honour of V.L. Selivanov's 60th Birthday, Ontos Verlag, De Gruyter Berlin,
Boston, 2014, pp. 922.
C.A. Agyingi, P. Haihambo and H.-P.A. Künzi, Endpoints in T0-quasi-metric spaces: Part
II, Abstract and Applied Analysis, Vol. 2013, Article ID 539573, 10 pages.
G. Berthiaume, On quasi-uniformities in hyperspaces, Proc. Amer. Math. Soc. 66 (1977),
335343.
M. Bukatin, R. Kopperman and S. Matthews, Some corollaries of the correspondence
between partial metrics and multivalued equalities, Fuzzy Sets Systems 256 (2014), 57
72.
M.J. Campión, E. Induráin, G. Ochoa and O. Valero, Functional equations related to
weighable quasi-metrics, Hacettepe J. Mat. Stat. 44 (4) (2015), 775787.
Ş. Cobzaş, Functional Analysis in Asymmetric Normed Spaces, Frontiers in Mathematics,
Springer, Basel, 2012.
P. Fletcher and W.F. Lindgren, Quasi-uniform Spaces, Dekker, New York, 1982.
Y.U. Gaba and H.-P.A. Künzi, Splitting metrics by T0-quasi-metrics, Topology Appl. 193
(2015), 8496.
Y.U. Gaba and H.-P.A. Künzi, Partially ordered metric spaces produced by T0-quasimetrics,
Topology Appl. 202 (2016), 366383.
J. Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory, Selected Topics in
Point-Set Topology, Cambridge University Press, Cambridge, 2013.
D. Gronau, A remark on Sincov's functional equation, Not. S. Afr. Math. Soc. 31 (2000),
1-8.
E. Kemajou, H.-P.A. Künzi and O.O. Otafudu, The Isbell-hull of a di-space, Topology
Appl. 159 (2012), 24632475.
H.-P.A. Künzi, An introduction to quasi-uniform spaces, Beyond Topology, Contemp.
Math. 486 (2009), 239304.
H.-P.A. Künzi and C. Makitu Kivuvu, A double completion for an arbitrary T0-quasimetric
space, J. Logic Algebraic Programming 76 (2008), 251269.
H.-P.A. Künzi and S. Romaguera, Weightable quasi-uniformities, Acta Math. Hungar.
136 (1-2), (2012), 107128.
H.-P.A. Künzi and C. Ryser, The Bourbaki quasi-uniformity, Topology Proceedings 20
(1995), 161183.
H.-P.A. Künzi and M. Sanchis, The Katetov construction modified for a T0-quasi-metric
space, Topology Appl. 159 (2012), 711720.
H.-P.A. Künzi and V. Vajner, Weighted quasi-metrics, Ann. New York Acad. Sci. 728
(1994), 6477.
H.-P. A. Künzi and F. Yıldız, Convexity structures in T0-quasi-metric spaces, Topology
Appl. 200 (2016), 218.
L. Nel, Continuity Theory, Springer, Switzerland, 2016.
F. Plastria, Asymmetric distances, semidirected networks and majority in Fermat-Weber
problems, Ann. Oper. Res. (2009) 167: 121155.
I.V. Protasov, Coronas of balleans, Topology Appl. 149 (2005), 149160.
T. Salát, J. Tóth and L. Zsilinszky, Metric space of metrics defined on a given set, Real
Anal Exch., 18 no. 1 (19921993), 225231.
T. Salát, J. Tóth and L. Zsilinszky, On the structure of the space of metrics defined on a
given set, Real Anal. Exch. 19, no. 1 (19931994), 321327.
Demetriou, N., & Künzi, H.-p. A. (2017). A study on quasi-pseudometrics. Hacettepe Journal of Mathematics and Statistics, 46(1), 33-52.
AMA
Demetriou N, Künzi HpA. A study on quasi-pseudometrics. Hacettepe Journal of Mathematics and Statistics. February 2017;46(1):33-52.
Chicago
Demetriou, Natasha, and Hans-peter A. Künzi. “A Study on Quasi-Pseudometrics”. Hacettepe Journal of Mathematics and Statistics 46, no. 1 (February 2017): 33-52.
EndNote
Demetriou N, Künzi H-pA (February 1, 2017) A study on quasi-pseudometrics. Hacettepe Journal of Mathematics and Statistics 46 1 33–52.
IEEE
N. Demetriou and H.-p. A. Künzi, “A study on quasi-pseudometrics”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 1, pp. 33–52, 2017.
ISNAD
Demetriou, Natasha - Künzi, Hans-peter A. “A Study on Quasi-Pseudometrics”. Hacettepe Journal of Mathematics and Statistics 46/1 (February 2017), 33-52.
JAMA
Demetriou N, Künzi H-pA. A study on quasi-pseudometrics. Hacettepe Journal of Mathematics and Statistics. 2017;46:33–52.
MLA
Demetriou, Natasha and Hans-peter A. Künzi. “A Study on Quasi-Pseudometrics”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 1, 2017, pp. 33-52.
Vancouver
Demetriou N, Künzi H-pA. A study on quasi-pseudometrics. Hacettepe Journal of Mathematics and Statistics. 2017;46(1):33-52.