Irrationals are Everywhere Dense in Reals

Theorem

Topology

Let $T = \struct {\R, \tau}$ denote the real number line with the usual (Euclidean) topology.

Let $\R \setminus \Q$ be the set of irrational numbers.

Then $\R \setminus \Q$ is everywhere dense in $T$.

Normed Vector Space

Let $\struct {\R, \size {\, \cdot \,} }$ be the normed vector space of real numbers.

Let $\R \setminus \Q$ be the set of irrational numbers.

Then $\R \setminus \Q$ is everywhere dense in $\struct {\R, \size {\, \cdot \,} }$