Let ${\cal H}$ be a Hilbert space (separable, if it matters), and let $X$ be a set of closed subspaces with the property that for all ${\cal P},{\cal Q}\in X$, we have either ${\cal P}\subset {\cal Q}$ or ${\cal Q}\subset {\cal P}$. We can assume that $X$ includes the trivial subspaces ${\cal H}$ and $\varnothing$ if that makes a difference. The set $X$ of closed subspaces is not necessarily countable.
Does such an $X$ have a concise name?
I'm not a mathematician. I tried searching for keywords like "sequence of subspaces", "nested subspaces", and "filter" (just guessing), but I didn't recognize anything relevant.
I also tried looking in the context of "resolution of the identity," which I suppose is what $X$ would be called if it were described in terms of projection operators instead of subspaces, but I didn't find any clear statements about whether or not that name still applies when $X$ is described in terms of closed subspaces.
Such a set $X$ is typically called a nest of subspaces. You could also call it a chain of subspaces: the word chain is used very broadly in mathematics to mean "totally ordered set", especially a totally ordered subset of a partially ordered set (what you are describing is exactly a totally ordered subset of the partially ordered set of closed subspaces).