Nowhere Dense iff Complement of Closure is Everywhere Dense/Warning

Nowhere Dense iff Complement of Closure is Everywhere Dense: Warning

Note that in the result:

$H$ is nowhere dense in $T$ if and only if the relative complement of its closure is everywhere dense in $T$

must be applied to the closure of $H$.

Otherwise, consider the real numbers $\R$ and the rational numbers $\Q$.

We have that:

$\R \setminus \Q$ is the set of irrational numbers.

We have that Irrationals are Everywhere Dense in Reals.

But we also have from Rationals are Everywhere Dense in Reals:

$\map \cl \Q = \R$

and so:

$\paren {\map \cl \Q}^\circ = \R$

So it is not the case that $\R \setminus \Q$ is nowhere dense in $\R$.

However:

$\R \setminus \map \cl \Q = \O$

which is indeed nowhere dense in $\R$.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Example $3.7.30$