Fences, their endpoints, and projective Fraïssé theory

Abstract:

We introduce a new class of compact metrizable spaces, which we call fences, and its subclass of smooth fences. We isolate two families $\mathcal {F}, \mathcal {F}_{0}$ of Hasse diagrams of finite partial orders and show that smooth fences are exactly the spaces which are approximated by projective sequences from $\mathcal {F}_{0}$. We investigate the combinatorial properties of Hasse diagrams of finite partial orders and show that $\mathcal {F}, \mathcal {F}_{0}$ are projective Fraïssé families with a common projective Fraïssé limit. We study this limit and characterize the smooth fence obtained as its quotient, which we call a Fraïssé fence. We show that the Fraïssé fence is a highly homogeneous space which shares several features with the Lelek fan, and we examine the structure of its spaces of endpoints. Along the way we establish some new facts in projective Fraïssé theory.