# Interior of Singleton in Real Number Space is Empty

## Theorem

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

Let $a \in \R$ be a real number.

Then:

$\set a^\circ = \O$

where $\set a^\circ$ denotes the interior of $\set a$ in $\R$.

## Proof

 $\displaystyle \set a^\circ$ $=$ $\displaystyle \closedint a a^\circ$ Definition of Closed Real Interval $\displaystyle$ $=$ $\displaystyle \openint a a$ Interior of Closed Real Interval is Open Real Interval $\displaystyle$ $=$ $\displaystyle \set {x \in \R: a < x < a}$ Definition of Open Real Interval $\displaystyle$ $=$ $\displaystyle \O$ Definition of Empty Real Interval

$\blacksquare$