Interior of Singleton in Real Number Space is Empty

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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

$\blacksquare$