Greshnov, A. V. Distance functions between sets in \((q_1, q_2)\)-quasimetric spaces. (English. Russian original) Zbl 1447.54021 Sib. Math. J. 61, No. 3, 417-425 (2020); translation from Sib. Mat. Zh. 61, No. 3, 528-538 (2020). Given a quasimetric space \((X, d)\), if the inequality \(d(x, y)\leq q_1d(x, z)+q_2d(z, y)\) holds for all \(x, y, z\in X\), then \((X, d)\) is called a \((q_1, q_2)\)-quasimetric space. In this paper the author studies the properties of some analogs of the Hausdorff distance in \((q_1, q_2)\)-quasimetric spaces. It is shown that if \((X, d)\) is a complete \((q_1, q_2)\)-quasimetric space and \(d\) is lower semicontinuous in the second argument, then \((\mathscr{M}(X), h)\) is complete, where \(\mathscr{M}(X)\) denotes the set of all \(d\)-closed \(d\)-bounded subsets of \(X\), and \(h\) is some analog of the Hausdorff \((q_1, q_2)\)-distance \(H\) on \(\mathscr{M}(X)\). Reviewer: Shou Lin (Ningde) Cited in 1 Document MSC: 54E35 Metric spaces, metrizability 54E50 Complete metric spaces 54B20 Hyperspaces in general topology Keywords:\((q_1, q_2)\)-quasimetric; Hausdorff distance; closed set; bounded set; completeness PDFBibTeX XMLCite \textit{A. V. Greshnov}, Sib. Math. J. 61, No. 3, 417--425 (2020; Zbl 1447.54021); translation from Sib. Mat. Zh. 61, No. 3, 528--538 (2020) Full Text: DOI References: [1] Wilson, W. A., On quasi-metric spaces, Amer. J. Math., 53, 3, 675-684 (1931) · Zbl 0002.05503 [2] Arutyunov, A. V.; Greshnov, A. V., (q_1, q_2)-Quasimetric spaces. Covering mappings and coincidence points, Izv. Math., 82, 2, 245-272 (2018) · Zbl 1401.54023 [3] Arutyunov, A. V.; Greshnov, A. V., Theory of (q_1, q_2)-quasimetric spaces and coincidence points, Dokl. Math., 94, 1, 434-437 (2016) · Zbl 1352.54030 [4] Arutyunov, A. V.; Greshnov, A. V., Coincidence points of multivalued mappings in (q_1, q_2)-quasimetric spaces, Dokl. Math., 96, 2, 438-441 (2017) · Zbl 1410.54045 [5] Gromov, M., Carnot-Carathéodory spaces seen from within, Sub-Riemannian Geometry, 79-323 (1996), Basel: Birkhäuser, Basel · Zbl 0864.53025 [6] Greshnov, A. V., Proof of Gromov’s theorem on homogeneous nilpotent approximation for vector fields of class C^1, Siberian Adv. Math., 23, 3, 180-191 (2013) · Zbl 1340.53064 [7] Greshnov, A. V., (q_1, q_2)-Quasimetrics bi-Lipschitz equivalent to 1-quasimetrics, Siberian Adv. Math., 27, 4, 253-262 (2017) · Zbl 1399.54061 [8] Burago, D. Yu; Burago, Yu D.; Ivanov, S. V., A Course in Metric Geometry (2001), Providence: Amer. Math. Soc., Providence · Zbl 0981.51016 [9] Arutyunov, A. V., Stability of coincidence points and properties of covering mappings, Math. Notes, 86, 2, 153-158 (2009) · Zbl 1186.54033 [10] Arutyunov, A. V., Covering mappings in metric spaces and fixed points, Dokl. Math., 76, 2, 665-668 (2007) · Zbl 1152.54351 [11] Arutyunov, A. V.; Greshnov, A. V.; Lokutsievskii, L. V.; Storozhuk, K. V., Topological and geometrical properties of spaces with symmetric and nonsymmetric f-quasimetrics, Topology Appl., 221, 178-194 (2017) · Zbl 1377.54028 [12] Cvetković M., Karapinar E., and Rakocević V., “Some fixed point results on quasi-b-metric-like spaces,” J. Inequal. Appl., no. 374, 17 pp. (2015). · Zbl 1347.54072 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.