Is such space monotonically monolithic or strongly monotonically monolithic? Let $R$ denote the set of all real numbers. $B$ is a subset of $R$ with $B=2^{\aleph_0}$. We topologize $R$ now: the set $B$ is discrete and its complement has the usual topology. So we can see $R$ is the union of a discrete subspace and metrizable space. My question is this: > Is such space monotonically monolithic or strongly monotonically monolithic? For the definitions of monotonically monolithic and strongly monotonically monolithic, see here.

Show that such a space has a point-countable base (like the Michael line; use a countable base for the usual topology on $R$ with all singletons from $B$), and all such spaces are monotonically monolithic, according to Tkachuk, "monolithic spaces and D-spaces revisited". The cardinality of $B$ is irrelevant.