On Generic Topological Embeddings

We show that an embedding of a fixed 0-dimensional compact space K into the Čech–Stone remainder $$\omega ^*$$ ω ∗ as a nowhere dense P-set is the unique generic limit, a special object in the category consisting of all continuous maps from K to compact metric spaces. Using categorical Fraïssé theory, we obtain some well-known theorems about Čech–Stone remainders of infinite discrete spaces. Furthermore, we prove several results concerning universality and homogeneity of the space $$\kappa ^\kappa $$ κ κ , with $$\kappa $$ κ a regular uncountable cardinal, using generalized ultrametrics.