Ytukyg

I am able to prove the iff in the forward direction. But, I am having trouble proving the statement in other direction. I am trying to use the definition of dense, but I am not getting anywhere with it.

Let $U subseteq X$ be an open set and let $S subseteq X$ be the set in question. Denseness is equivalent to saying that it intersects every open set non-trivially, hence $U cap bar{S}^c neq emptyset$. Considering properties of the complement, this is the same as saying $U$ is not a subset of $bar{S}$. Since the closure of $S$ does not contain an open set, it has empty interior and so $S$ is nowhere dense.

Let $S^{c}$ denote the complement of $S$ in $X$ and $cl(S)$ the closure of $S$ in $X$.

To prove that $S$ is nowhere dense, any open set $U subseteq cl(S)$ must be empty. Let $U$ be such a set.

The complements are included in the opposite order, that is $cl(S)^{c} subseteq U^{c}$. Taking the closures of both sides and noting that $U^{c}$ is closed, this gives $cl( cl(S)^{c})) subseteq U^{c}$.

But $cl(S)^{c}$ is by assumption dense, which (by definition) means that $cl( cl(S)^{c})) = X$. We thus got ourselves a relation $X subseteq U^{c}$. This is possible only if $U = emptyset$.

Hence any open subset of $cl(S)$ must be empty and thus $S$ is nowhere dense.

Let $A^c$ be complement of $A$, $overline{A}$ be closure of $A$ and $A^o$ be interior of $A$.

We define:

1). $A$ is dense iff $overline{A}=X$.

2). $A$ is nowhere dense iff $(overline{A})^o=varnothing$.

Edit:

We will use the facts that $(A^c)^c=A$ (Involution) in the following proof. First we prove following lemma
$$
overline{A}=((A^c)^o)^ctag1
$$
By definition, $A^o$ (interior set of $A$) is the largest open contained in $A$. So $(A^c)^o$ is the largest open contained in $A^c$, i.e $(A^c)^osubset A^c$. Thus $A=(A^c)^csubset ((A^c)^o)^c$. Since $(A^c)^o$ is open, $((A^c)^o)^c$ is closed. So this means that $((A^c)^o)^c$ is the smallest closed set containing $A$. By definition of closure, $(1)$ follows.

Now if $A$ be nowhere dense, then $(overline{A})^o=varnothing$. Since by $(1)$ and Involution
$$
overline{(overline{A})^c}=((((overline{A})^{c})^c)^o)^c=((overline{A})^o)^c=varnothing^c=X
$$
i.e. $(overline{A})^c$ is dense.

Second if $(overline{A})^c$ is dense, then $overline{(overline{A})^c}=X$ and
$$
((overline{A})^o)^c=((((overline{A})^{c})^c)^o)^c=overline{(overline{A})^c}=X
$$
So $(overline{A})^o=varnothing$, i.e $A$ is nowhere dense.

Assume that complement of (clA) is dense.
Then cl[complement of (clA)] = X.
So complement of (int clA) = X.
Thus int clA is empty.
Hence A is nowhere dense.