Abstract
We prove theorems on the topological equivalence of distance functions on spaces with weak and reverse weak symmetries. We study the topology induced by a distance function ρ under the condition of the existence of a lower symmetrization for ρ by an f-quasimetric. For (q1, q2)-metric spaces (X, ρ), we also study the properties of their symmetrizations min {ρ(x, y), ρ(y, x)} and max {ρ(x, y), ρ(y, x)}. The relationship between the extreme points of a (q1q2)-quasimetric ρ and its symmetrizations min{ρ(x, y), ρ(y, x)} and max {ρ(x, y), ρ(y, x)}.
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References
H. Aimar, L. Forzani, and R. Toledano, “Balls and quasi-metrics: a space of homogeneous type modeling the real analysis related to the Monge-Ampére equation,” J. Fourier Anal. Appl. 4 (4–5), 377 (1998).
A. V. Arutyunov and A. V. Greshnov, “Theory of (q 1, q 2)-quasimetric spaces and coincidence points,” Dokl. Akad. Nauk 469, 527 (2016) [Dokl. Math. 94, 434 (2016)].
A. V. Arutyunov and A. V. Greshnov, “(q 1, q 2)-quasimetric spaces. Covering mappings and coincidence points,” Izv. Ross. Akad. Nauk, Ser. Mat. 82 (2), 3 (2018) [Izv. Math. 82, 245 (2018)]
A. V. Arutyunov, A. V. Greshnov, L. V. Lokutsievskiĭ, and K. V. Storozhuk, “Topological and geometrical properties of spaces with symmetric and nonsymmetric f-quasimetrics,” Topology Appl. 221, 178 (2017).
M. Balanzat, “Sobre la metrización de los espacios cuasi métricos,” Gaz. Mat. Lisboa 50, 90 (1951).
A. V. Greshnov, “Proof of Gromov’s theorem on homogeneous nilpotent approximation for vector fields of class C 1, Mat. Tr. 15 (2), 72 (2012) [Sib. Adv. Math. 23, 180 (2013)].
A. V. Greshnov, “(q 1, q 2)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics,” Mat. Tr. 20 (1), 81 (2017) [Sib. Adv. Math. 27, 253 (2017)].
A. V. Greshnov, “Regularization of distance functions and separation axioms on (q 1 q 2)-quasimetric spaces,” Sib. Èlektron. Mat. Izv. 14, 765 (2017).
J. Heinonen, Lectures on Analysis on Metric Spaces (Springer-Verlag, New York, 2001).
M. Karmanova and S. Vodop′yanov, “Geometry of Carnot-Carathéodory spaces, differentiability, coarea and area formulas,” Analysis and Mathematical Physics. (Trends Math.), 233 (Birkhäuser, Basel, 2009).
S. Ĭ. Nedev, “o-metrizable spaces,” Tr. Mosk. Mat. Obs., 24, 201 (1971) [Trans. Mosc. Math. Soc. 24, 213 (1974)].
R. Sengupta, “On fixed points of contraction mappings acting in (q 1 q 2)-quasimetric spaces and geometric properties of these spaces,” Eurasian Math. J. 8 (3), 70 (2017).
W. A. Wilson, “On quasi-metric spaces,” Amer. J. Math. 53, 675 (1931).
Acknowledgments
The author expresses his deep gratitude to the referee for evincing interest in his work.
Funding
The work was supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant 1.1.2; Project 0314-2016-0006).
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Russian Text © The Author(s), 2018, published in Matematicheskie Trudy, 2018, Vol. 21, No. 2, pp. 150–162.
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Greshnov, A.V. Symmetrizations of Distance Functions and f-Quasimetric Spaces. Sib. Adv. Math. 29, 202–209 (2019). https://doi.org/10.3103/S1055134419030052
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DOI: https://doi.org/10.3103/S1055134419030052