Let $(X, \mathfrak{T})$ be a topological space
Let $V, U \in \mathfrak{T}$, and suppose that $V \subseteq U$
Then when it is true that $V \subseteq \overline V \subseteq U$, where $\overline V$ is the closure of $V$?
(Properties of the underlying topology for example)
This doesn't seem to be true in general, consider the particular point topology
$(X, \mathfrak{T}_p)$
Let $V, U \in \mathfrak{T}_p$, $V \subseteq U$ , then $p \in V, U$, but $\overline V = X$