analog of the complete Erdős space.

To obtain our results, we establish combinatorial criteria which are of general interest in the context of projective [Fraïssé ]{}theory. characterizes which projective sequences of structures in a language containing a binary relation symbol $\set{R}$ have limit on which $R$ is an equivalence relation, and gives conditions under which the resulting quotient map is irreducible. The irreducibility condition entails a correspondence between structures in the projective sequence and regular quasi-partitions of the quotient, which in turn aids the combinatorial-topological translation.

Here is the plan of the paper. We begin in with recalling some notions and proving some technical lemmas which will lay the basis of this work. In we introduce the topological structures that constitute the main combinatorial objects of our study, prove that the relevant classes $ {\mathcal F}$ and $ {{\mathcal F}_{0}}$ are projective Fraïssé and investigate the properties of the projective limits of $ {{\mathcal F}_{0}}$. We define fences and characterize smooth fences in , where we also display the relation linking them to $ {{\mathcal F}_{0}}$. Finally in we characterize topologically the quotient of the projective [Fraïssé ]{}limit of ${\mathcal F}$, explore its homogeneity features and investigate its spaces of 