Interior of Set of Rational Numbers in Real Numbers is Empty

Theorem

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\Q$ be the subspace of rational numbers.

Then the interior of $\Q$ in $\R$ is the empty set $\O$.

Proof

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

By definition:

$\R \setminus \Q = \relcomp \R \Q$

where $\relcomp \R \Q$ denotes the relative complement of $\Q$ in $\R$.

We have that Irrationals are Everywhere Dense in Reals.

Hence by definition of everywhere dense, the closure of $\R \setminus \Q$ in $\R$ is $\R$ itself.

By Relative Complement with Self is Empty Set:

$\relcomp \R \R = \O$

The result follows from Interior equals Complement of Closure of Complement.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Examples $3.7.25$