On generic topological embeddings

Abstract

We show that an embedding of a fixed 0-dimensional compact space $K$ into the Čech--Stone remainder $\ddot{I}{‰}^{\ast }$ 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 $\hat{I}º$ can be uniformly embedded into $\hat{I}{º}^{\hat{I}}º$ as a uniformly nowhere dense subset, -- every uniform homeomorphism of uniformly nowhere dense sets in $\hat{I}{º}^{\hat{I}}º$ can be extended to a uniform auto-homeomorphism of $\hat{I}{º}^{\hat{I}}º$, -- every uniformly nowhere dense set in $\hat{I}{º}^{\hat{I}}º$ is a uniform retract of $\hat{I}{º}^{\hat{I}}º$. If we assume that $\hat{I}º$ is a weakly compact cardinal we get the counterpart of the above result without the uniformity assumption.