Interior of Singleton in Real Number Line is Empty

Theorem

Let $\struct {\R, \tau_d}$ be the real number line with 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

\(\ds \set a^\circ\)

\(=\)

\(\ds \closedint a a^\circ\)

Definition of Closed Real Interval

\(\ds \)

\(=\)

\(\ds \openint a a\)

Interior of Closed Real Interval is Open Real Interval

\(\ds \)

\(=\)

\(\ds \set {x \in \R: a < x < a}\)

Definition of Open Real Interval

\(\ds \)

\(=\)

\(\ds \O\)

Definition of Empty Real Interval

$\blacksquare$