> A non-empty complete metric space is NOT the countable union of nowhere-dense closed sets.
I don't understand the following point the proof: we assume that $X$ is the countable union of nowhere-dense closed sets. $$ X=\bigcup_{n=1}^\infty C_n $$ and we can choose $x_1 \in A_1=(C_1)^c \,\Rightarrow\exists\,\, \varepsilon_1 <1 : \overline{B(x_1,\varepsilon_1)}\subset A_1$ because $C_1$ has no interior points. What I don't get is the last "because". Someone can help me in understanding this passage of the proof? Fortunately, the rest of the proof is clear to me.