Recognizing the topologies of spaces of metrics with the topology of uniform convergence

• Autorzy: Koshino Katsuhisa

Identyfikatory

DOI

10.4064/ba220523-18-4

• Języki publikacji: EN

Abstrakty

EN

Given a metrizable space X of density κ, we study the topological structure of the space PM(X) of continuous bounded pseudometrics on X, which is endowed with the topology of uniform convergence. We prove that PM(X) is homeomorphic to [0,1)κ(κ−1)/2 if X is finite, to ℓ2(2<κ) if X is infinite and generalized compact, and to ℓ2(2κ) if X is not generalized compact. We also show that for an infinite σ-compact metrizable space X, the space M(X)⊂PM(X) of continuous bounded metrics on X and the space AM(X)⊂M(X) of bounded admissible metrics on X are homeomorphic to ℓ2 if X is compact, and to ℓ∞ if X is not compact.

Słowa kluczowe

EN

pseudometric; admissible metric; topology of uniform convergence; sup-metric; Fréchet space; convex set; Hilbert space

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2022
• Tom: Vol. 70, no. 2
• Strony: 165--171
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Koshino Katsuhisa

• Faculty of Engineering, Kanagawa University, Yokohama, 221-8686, Japan

Bibliografia

• [1] A. Barbati and C. Costantini, On the density of the hyperspace of a metric space, Comment. Math. Univ. Carolin. 38 (1997), 349-360.
• [2] T. Banakh and R. Cauty, Topological classification of closed convex sets in Fréchet spaces, Studia Math. 205 (2011), 1-11.
• [3] C. Costantini, On the density of the space of continuous and uniformly continuous functions, Topology Appl. 153 (2006), 1056-1078.
• [4] T. Dobrowolski and H. Toruńczyk, Separable complete ANR’s admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), 229-235.
• [5] F. Hausdorff, Erweiterung einer Homöomorphie, Fund. Math. 16 (1930), 353-360.
• [6] Y. Ishiki, An interpolation of metrics and spaces of metrics, arXiv:2003.13227 (2020).
• [7] K. Koshino, The topological type of spaces consisting of certain metrics on locally compact metrizable spaces with the compact-open topology, arXiv:2202.08615 (2022).
• [8] K. Sakai, Geometric Aspects of General Topology, Springer Monogr. Math., Springer, Tokyo, 2013.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).