If A is a subset of Metric Space M such that A is nowhere dense , then Clo(A) contains no open set of M

Question

I am not understanding this def. of Nowhere dense set in a metric space. Though I know another form of def. which states that a subset A of Metric Space M is nowhere dense iff, Int(Clo A)= Null set.

Answer 1

Argue by contradiction. Assume that $A$ contains a non-empty open set of $M$, call it $U \subset A$. Let $x \in U$ be any point. Clearly, $x$ is an interior point of $U$, so there exists an open sphere $S(x,r)$ centered at $x$ and contained in $U$.

Since $U$ is contained in $A$, $S(x,r)$ is contained in $A$.

Since $A$ is contained in $\mbox{cl } A$, $S(x,r)$ is contained in $\mbox{cl } A$. We get that there is a point $x \in U \subset \mbox{cl } A$ such that $S(x,r)$ is contained in $\mbox{cl } A$.

Then, $x$ is an interior point of $\mbox{cl } A$, a contradiction as the interior of the closure is given to be empty.