On generic topological embeddings

We show that an embedding of a fixed 0-dimensional compact space $K$ into the Čech–Stone remainder ${\omega }^{\ast }$ 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 Fraïssé theory we get a few well know theorems about Čech–Stone remainder. We establish the following: – an ultrametric space $K$ of weight $\kappa$ can be uniformly embedded into ${\kappa }^{\kappa }$ as a uniformly nowhere dense subset, – every uniform homeomorphism of uniformly nowhere dense sets in ${\kappa }^{\kappa }$ can be extended to a uniform auto-homeomorphism of ${\kappa }^{\kappa }$, – every uniformly nowhere dense set in ${\kappa }^{\kappa }$ is a uniform retract of ${\kappa }^{\kappa }$. If we assume that $\kappa$ is a weakly compact cardinal we get the counterpart of the above result without the uniformity assumption.