Exercise 7, § 1

Consider the following four properties of a given topological space $X$:

(D_I) The topology of $X$ has a countable base.

(D_{II}) $X$ has a countable dense subset.

(D_{III}) Every subset of $X$, all of whose points are isolated, is countable.

(D_{IV}) Every set of mutually disjoint non-empty open subsets of $X$ is countable.

Show that $(D_I)$ implies $(D_{II})$ and $(D_{III})$, and that each of $(D_{II})$ and $(D_{III})$ imply $(D_{IV})$ (*).