Relative (functionally) Type I spaces and narrow subspaces

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 $\kappa=\omega_1$ and the closure of each $U_\alpha$ is Lindel\"of. A closed subspace $D\subset X$ is narrow in $X$ if for each systematic cover $\{V_\alpha : \alpha\in\omega_1\}$ of $X$, either there is $\alpha$ such that $D$ is included in $V_\alpha$, or the closure of $V_\alpha$ intersected with $D$ is Lindel\"of for each $\alpha$. Taking systematic covers given by preimages by $s$ of $[0,\alpha)$ for a continuous $s: X\to \mathbb{L}_{\ge 0}$ (where $\mathbb{L}_{\ge 0}$ is the longray) defines functionally Type I spaces and functionally narrow subspaces. For instance, $\mathbb{L}_{\ge 0}$ and $\omega_1$ 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 $\omega_1$-compact locally compact countably tight Type I space contains a non-Lindel\"of subspace narrow in it (a copy of $\omega_1$, 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 $\omega$- or $\omega_1$-closed.