Subset of Nowhere Dense Subset is Nowhere Dense

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$ be nowhere dense in $T$.

Let $B \subseteq A$.

Then $B$ is nowhere dense in $T$.

Proof
Aiming for a proof by contrapositive, suppose that $B$ is not nowhere dense in $T$.

Then by definition 2 of nowhere dense:


 * $B^-$ contains some open set of $T$ which is non-empty.

From Set Closure Preserves Set Inclusion, we have:


 * $B^- \subseteq A^-$

So:


 * $A^-$ contains some open set of $T$ which is non-empty.

So $A$ is not nowhere dense in $T$.

Hence by the Rule of Transposition:


 * if $A$ is nowhere dense in $T$ then $B$ must be nowhere dense in $T$.