A combinatorial model for the Menger curve

Abstract

We represent the universal Menger curve as the topological realization |M| of the projective Fraïssé limit M of the class of all finite connected graphs. We show that 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 M → |M|, we transfer some of these properties to the Menger curve: we prove the approximate projective homogeneity theorem, recover Anderson's finite homogeneity theorem, and prove a variant of Anderson-Wilson's theorem. The finite homogeneity theorem is the first instance of an "injective" homogeneity theorem being proved using the projective Fraïssé method. We indicate how our approach to the Menger curve may extend to higher dimensions.2010 Mathematics Subject Classification. 03C30, 54F15.

The homeomorphism group of the universal Knaster continuum

“…Following [14] and [9], we say that a class F of finite L-structures with a fixed family of morphisms is a projective Fraissé family if:…”

“…Enlarging category K. We first must define an enlargement of the category K to include possible limit objects. This type of enlargement is used in [14] and we follow the development from that paper. A topological pointed graph, (X, R X , c X ), consists of a compact, metrizable, zero-dimensional domain space, X; a reflexsive, and symmetric relation R X ⊆ X 2 which is a closed subset of X 2 ; and a designated point c X ∈ X. Epimorphisms between topological pointed graphs must take edges to edges, preserve the designated point, be surjective on vertives and edges, and moreover must be continuous.…”

“…In the dual setting, the limit satisfies analogous properties with the arrow directions "swapped." We will follow [9] and [14] for the development of projective Fraissé limits.…”

“…Theorem 4.2 (Theorem 3.1 of [14]). If F is a projective Fraissé family of finite (pointed) graphs, then there exists a unique up to isomorphism (pointed) topological graph F ∈ F ω so that (1) for each A ∈ F, there is a morphism in F ω from F → A (2) for A, B ∈ F and morphisms f :…”

Hereditarily indecomposable continua as generic objects

“…Here the term "pre-space" means the Cantor space together with a special closed equivalence relation. Since then, many other metrizable compacta were realized as quotients of pre-spaces that are projective Fraïssé limits, for example the Lelek fan by Bartošová and Kwiatkowska [3], the Menger curve by Panagiotopoulos and Solecki [39], the so called Fraïssé fence by Basso and Camerlo [4], and the Cantor fan and the generalized Ważewski dendrite D 3 by Charatonik and Roe [10]. On the other hand, the second author [27] introduced a framework for approximate Fraïssé theory based on metric-enriched categories and realized the pseudo-arc as a Fraïssé limit directly, i.e.…”