Exercise 1, § 5

a) If $A$ is a subset of a topological space $X$, show that the following statements are equivalent: $\alpha$ the frontier of $A$ is nowhere dense; $\beta$ $A$ is the union of an open set and a nowhere dense set; $\gamma$ $A$ is the difference between a closed set and a nowhere dense set.

$b$ Show that if a subset $A$ of $X$ is such that, for each point $x \in A$, there is a neighbourhood $V$ of $x$ in $X$ such that $V \cap A$ is nowhere dense in $X$, then $A$ is nowhere dense in $X$.