Relative (Functionally) Type I Spaces and Narrow Subspaces - Baillif, Mathieu (Topology Proceedings, 2023)
An open chain cover U={U_α: α κ} ( κ a cardinal) of a space X is a systematic cover if U_α U_β when α<β, and X is Type I if κ=ω_1 and each U_α is Lindelöf. A closed subspace D X is narrow in X if for each systematic cover {V_α: α ω_1} of X, either there is α such that D V_α or V_α D is Lindelöf for each α. Taking systematic covers given by s^-1([0, α)) for a continuous s : X → L_≥ 0 (where L_≥ 0 is the long ray) defines functionally Type I spaces and functionally narrow subspaces. For inst ance, L_≥ 0 and ω_1 are narrow in themselves and any other space. We investigate these properties and relative versions, as well as their relationship, and show in particular the following. There are functionally Hausdorff Type I spaces which are not functionally Type I, while regular Type I spaces are functionally Type I. We exhibit examples of spaces which are narrow in some but not in other spaces. There are subspaces of a Tychonoff space Y that are functionally narrow but not narrow in Y, while both notions agree if Y is normal. Under PFA and using classical results, any ω_1-compact locally compact countably tight Type I space contains a non-Lindelöf subspace narrow in it (a copy of ω_1, actually), while a Suslin tree does not. There are spaces with subspaces narrow in them that are essentially discrete. Finally, we investigate natural partial orders on (functionally) narrow subspaces and when these orders are ω - or ω_1-closed.