Basic question about open and closed sets Let $A$ and $B$ be open sets in a metric space $X$ and suppose that $B\subseteq \overline{A}$ (where $\overline{A}$ denotes the closure of $A$). Is it true that $B\subseteq A$ ?

Minimal counterexample. Take the Sierpiński space. It is the $2$-element set $X = \{0, 1\}$ whose open sets are $\emptyset$, $\{1\}$ and $X$. Now take $A = \{1\}$ and $B = X$. Then $A$ and $B$ are open and $B \subseteq \overline{A}$ (since $\overline{A} = X$), but $B \not\subseteq A$.
