Uniform homogeneity

Uniqueness, universality, and homogeneity of the noncommutative Gurarij space

We realize the noncommutative Gurarij space $\mathbb{NG}$ defined by Oikhberg as the Fra\"ıssé limit of the class of finite-dimensional $1$-exact operator spaces. As a consequence we deduce that the concommutative Gurarij space is unique up to completely isometric isomorphism, homogeneous, and universal among separable $1$-exact operator spaces. We also prove that $\mathbb{NG}$ is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal TRO to the triple envelope is an isomorphism. We deduce from this fact that $\mathbb{NG}$ does not e…

Homogeneity of abstract linear spaces

We discuss homogeneity and universality issues in the theory of abstract linear spaces, namely, structures with points and lines satisfying natural axioms, as in Euclidean or projective geometry. We show that the two smallest projective planes (including the Fano plane) are homogeneous and, assuming the continuum hypothesis, there exists a universal projective plane of cardinality $\aleph_1$ that is homogeneous with respect to its countable and finite projective subplanes. We also show that the existence of a generic countable linear space is equivalent to an old conjecture asserting that ever…

A combinatorial model for the Menger curve

We represent the universal Menger curve as the topological realization $|\mathbb{M}|$ of the projective Fraïssé limit ${\mathbb M}$ of the class of all finite connected graphs. We show that $\mathbb{M}$ satisfies combinatorial analogues of the Mayer-Oversteegen-Tymchatyn homogeneity theorem and the Anderson-Wilson projective universality theorem. Our arguments involve only $0$-dimensional topology and constructions on finite graphs. Using the topological realization $\mathbb{M}\mapsto|\mathbb{M}|$, we transfer some of these properties to the Menger curve: we prove the approximate projective ho…

齐性超度量结构

Homogeneous ultrametric structures

该研究以有限结构的Fraïssé类为起点、泛齐性满同态为核心工具，建立了具有严格齐性的齐性波兰超度量结构理论，将其作为Fraïssé极限类Borel层级构建的第三步。