In each dimension $d$, there exists a canonical compact, second countable space, called the $d$-dimensional Menger space, with certain universality and homogeneity properties. For $d = 0$, it is the Cantor set, for $d = \infty$, it is the Hilbert cube. In this talk, I will concentrate on the 1-dimensional Menger space. I will prove that it is a quotient of a projective Fraisse limit (a construction with roots in Model Theory). I will describe how a property of projective Fraisse limits coming from Logic, called the projective extension property, can be used to prove high homogeneity of the 1-dimensional Menger space.
This is joint work with Aristotelis Panagiotopoulos.