In this paper, the Urysohn, completely Hausdorff and completely regular axioms in $L$-topological spaces are generalized to $L$-fuzzy topological spaces. Each $L$-fuzzy topological space can be regarded to be Urysohn, completely Hausdorff and completely regular tosome degree. Some properties of them are investigated. The relations among them and $T_2$ in $L$-fuzzy topological spaces are discussed.

In a paper published posthumously, Pavel Samuilovich Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space. In this paper we prove that the Urysohn univeral space is hyperconvex.

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for ℓ 2 in the context of the Urysohn space U. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.

In this informal note, we demonstrate the existence of forking and nondividing formulas in continuous theory of the complete Urysohn sphere, as well as the discrete theories of the integral Urysohn spaces of diameter n (where n ≥ 3). Whether or not such formulas existed was asked in thesis work of the author, as well as joint work with Terry. We also show an interesting phenomenon in that, for ...

We solve the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for `2 in the context of the Urysohn space U. This is achieved by solving a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many dis-

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for ℓ 2 in the context of the Urysohn space U. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.

We study the oscillation stability problem for the Urysohn sphere, an analog of the distortion problem for ℓ 2 in the context of the Urysohn space U. In particular, we show that this problem reduces to a purely combinatorial problem involving a family of countable ultrahomogeneous metric spaces with finitely many distances.

We discuss some basic geometry of sets definable in the Urysohn sphere using only finitely many parameters and briefly remark on the case of arbitrary definable sets. Then we discuss definable functions in the Urysohn sphere satisfying a special syntactic property.

We construct the Urysohn metric space in constructive setting without choice principles. The Urysohn space is a complete separable metric space which contains an isometric copy of every separable metric space, and any isometric embedding into it from a finite subspace of a separable metric space extends to the whole domain.