Constructions of Urysohn universal ultrametric spaces

Abstract

In this paper, we give new constructions of Urysohn universal ultrametric spaces. We first characterize a Urysohn universal ultrametric subspace of the space of all continuous functions whose images contain the zero, from a zero-dimensional compact Hausdorff space without isolated points into the space of nonnegative real numbers equipped with the nearly discrete topology. As a consequence, the whole function space is Urysohn universal, which can be considered as a non-Archimedean analog of Banach-Mazur theorem. As a more application, we prove that the space of all continuous pseudo-ultrametrics on a zero-dimensional compact Hausdorff space with an accumulation point is a Urysohn universal ultrametric space. This result can be considered as a variant of Wan's construction of Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametric space.

Uniqueness and homogeneity of non-separable Urysohn universal ultrametric spaces

“…In the next theorem, we show that injective spaces constructed in [5], [13], and [2] become naturally petaloid. The proof and the precise definition of the spaces will be given in Section 4.…”

“…For d, e ∈ Cpu(X, S), we define UD S X (d, e) the infimum of all ǫ ∈ S such that d(x, y) ≤ e(x, y) ∨ ǫ and e(x, y) ≤ d(x, y) ∨ ǫ for all x, y ∈ X. For more details on C 0 (X, R) and Cpu(X, R), we refer the readers to [5].…”

“…This isometry theorem is proven by the so-called back-and-forth argument, which is a variant of the mathematical induction. Some authors have recently investigated a non-Archimedean analogue of the Urysohn universal space (see [5], [12], [2], and [13]), which is also our main subject of the paper.…”

“…For a range set R, there are several constructions of complete N(R)injective R-valued ultrametric spaces. For example, in [5], the author provided some new constructions of complete N(R)-injective Rultrametric space (see also [12], [2], and [13]). If R is countable, then, by the back-and-forth argument, a separable complete N(R)-injective R-ultrametric space is unique up to isometry whichever construction we choose.…”

“…) is said to be tenuous if it is finite or semi-sporadic (see [5]). For a range set R, we denote by TEN(R) the set of all tenuous range subsets of R. In this paper, we often represent a restricted metric as the same symbol to the ambient one.…”