Relative (functionally) Type I spaces and narrow subspaces

Mathieu Baillif

An open chain cover $\{U_{\alpha }:\alpha \in \kappa \}$ ($\kappa$ a cardinal) of a space $X$ is a systematic cover if the closure of $U_{\alpha }$ is contained in $U_{\beta }$ when $\alpha <\beta$, and $X$ is Type I if $\kappa ={\omega }_1$ an…