Strongly rigid metrics in spaces of metrics
Keywords:
Strong rigidity, Rigidity, Space of metrics, Comeagerness, Baire spaceAbstract
A metric space is said to be strongly rigid if no positive distance is taken twice by the metric. In 1972, Janos proved that a separable metrizable space has a strongly rigid metric if and only if it is zero-dimensional. In this paper, we shall develop this result for the theory of spaces of metrics. For a strongly zero-dimensional metrizable space, we prove that the set of all strongly rigid metrics is dense in the space of metrics. Moreover, if the space is the union of countably many compact subspaces, then that set is comeager. As a consequence, we show that for a strongly zero-dimensional metrizable space, the set of all metrics possessing no nontrivial (bijective) self-isometry is comeager in the space of metrics.
References
K. A. Broughan. A metric characterizing Cech dimension zero. Proc. Amer. Math. Soc., 39:437-440, 1973.
D. G. Ebin. The manifold of Riemannian metrics. In Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), pages 11-40. Amer. Math. Soc., Providence, R.I., 1970.
R. L. Ellis. Extending continuous functions on zero-dimensional spaces. Math. Ann., 186(2):114-122, 1970.
Y. Hattori. Congruence and dimension of nonseparable metric spaces. Proc. Amer. Math. Soc., 108(4):1103-1105, 1990.
C. Ho and S. Zimmerman. Partitioning the real line into an uncountable collection of everywhere uncountably dense sets. Amer. Math. Monthly, 126(9):825-834, 2019.
F. G. Dorais (https://mathoverflow.net/users/2000/françois-g dorais). explicit big linearly independent sets. MathOverflow. URL:https://mathoverflow.net/q/23206 (version: 2017-04-13).
Y. Ishiki. An interpolation of metrics and spaces of metrics. 2020. preprint, arXiv:2003.13277.
Y. Ishiki. An embedding, an extension, and an interpolation of ultrametrics. $p$-Adic Numbers Ultrametric Anal. Appl., 13(2):117-147, 2021.
Y. Ishiki. On dense subsets in spaces of metrics. 2021. preprint arXiv:2104.12450, to apper in Colloq. Math.
Y. Ishiki.Extending proper metrics. 2022. preprint arXiv:2207.12905, to appear in Topology Appl.
Y. Ishiki. On comeager sets of metrics whose ranges are disconnected. 2022. preprint arXiv:2207.12765, to appear in Topology Appl.
L. Janos. A metric characterization of zero-dimensional spaces. Proc. Amer. Math. Soc., 31:268-270, 1972.
L. Janos and H. Martin. Metric characterizations of dimension for separable metric spaces. Proc. Amer. Math. Soc., 70(2):209-212, 1978.
T. Jech. Set theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded.
H. W. Martin. Strongly rigid metrics and zero dimensionality. Proc. Amer. Math. Soc., 67 (1):157-161, 1977.
P. Mounoud. Metrics without isometries are generic. Monatsh. Math., 176(4):603-606, 2015.
J. Mycielski. Independent sets in topological algebras. Fund. Math., 55:139-147, 1964.
J. v. Neumann. Ein System algebraisch unabhängiger Zahlen. Math. Ann., 99(1):134-141, 1928.
R. S. Palais. When proper maps are closed. Proc. Amer. Math. Soc., 24:835-836, 1970.
A. R. Pears. Dimension theory of general spaces. Cambridge University Press, Cambridge, England-New York-Melbourne, 1975.
J. Rouyer. Generic properties of compact metric spaces. Topology Appl., 158(16):2140-2147, 2011.
P. Roy. Failure of equivalence of dimension concepts for metric spaces. Bull. Amer. Math. Soc., 68:609-613, 1962.
P. Roy. Nonequality of dimensions for metric spaces. Trans. Amer. Math. Soc., 134:117-132, 1968.
A. H. Stone. Non-separable Borel sets. Rozprawy Mat., 28:41, 1962.
