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 = infinity, it is the Hilbert cube. I will concentrate on the 1-dimensional Menger space. I will prove that it is a quotient of a projective Fraisse limit. 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.

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