Some aspects of Isbell-convex quasi-metric spaces

Olivier Olela Otafudu

doi:10.4995/agt.2018.7997

Authors

Olivier Olela Otafudu University of the Witwatersrand

DOI:

https://doi.org/10.4995/agt.2018.7997

Keywords:

Isbell-convexity, geodesic bicombing, injectivity

Abstract

We introduce and investigate the concept of geodesic bicombing in T₀-quasi-metric spaces. We prove that any Isbell-convex T₀-quasi-metric space admits a geodesic bicombing which satisfies the equivariance property. Additionally, we show that many results on geodesic bicombing hold in quasi-metric settings, provided that non symmetry in quasi-metric spaces holds.

Downloads

Download data is not yet available.

Author Biography

Olivier Olela Otafudu, University of the Witwatersrand

School of Mathematics

References

C. A. Agyingi, P. Haihambo and H.-P. A. Künzi, Tight extensions of T0-quasi-metric spaces, Logic, computation, hierarchies, Ontos Math. Log., 4, De Gruyter, Berlin, 2014, pp 9-22.

G. Berthiaume, On quasi-uniformities in hyperspaces, Proc. Amer. Math. Soc. 66 (1977), 335-343. https://doi.org/10.1090/S0002-9939-1977-0482620-9

J. Conradie, H.-P. A. ünzi and O. Olela Otafudu, The vector lattice structure on the Isbell-convex hull of an asymmetrically normed real vector space, Topology Appl. 231 (2017), 92-112. https://doi.org/10.1016/j.topol.2017.09.005

D. Descombes and U. Lang, Convex geodesic bicombings and hyperbolicity, Geom. Dedicata 177 (2015), 367-384. https://doi.org/10.1007/s10711-014-9994-y

A. W. M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. Math. 53 (1984), 321-402. https://doi.org/10.1016/0001-8708(84)90029-X

J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65-76. https://doi.org/10.1007/BF02566944

E. Kemajou, H.-P.A. Künzi and O. Olela Otafudu, The Isbell-hull of a di-space, Topology Appl. 159 (2012), 2463-2475. https://doi.org/10.1016/j.topol.2011.02.016

H.-P. A. Künzi and C. Ryser, The Bourbaki quasi-uniformity, Topology Proc. 20 (1995), 161-183.

H.-P. A. Künzi and F. Yildiz, Convexity structures in T_0-quasi-metric spaces, Topology Appl. 200 (2016), 2-18. https://doi.org/10.1016/j.topol.2015.12.009

U. Lang. Injective hulls of certain discrete metric spaces and groups. J. Topol. Anal. 5 (2013) 297-331. https://doi.org/10.1142/S1793525313500118

O. Olela Otafudu and Z. Mushaandja, Versatile asymmetrical tight extensions, Topol. Algebra Appl. 5 (2017), 6-12 https://doi.org/10.1515/taa-2017-0002