Open Real Interval is Regular Open

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Let $\left({a \,.\,.\, b}\right)$ be an open interval of $\R$.


Then $\left({a \,.\,.\, b}\right)$ is regular open in $\left({\R, \tau_d}\right)$.


Proof

From Open Sets in Real Number Line, $\left({a \,.\,.\, b}\right)$ is open in $\left({\R, \tau_d}\right)$.

From Closure of Open Real Interval is Closed Real Interval:

$\left({a \,.\,.\, b}\right)^- = \left[{a \,.\,.\, b}\right]$

where $\left({a \,.\,.\, b}\right)^-$ denotes the closure of $\left({a \,.\,.\, b}\right)$.

From Interior of Closed Real Interval is Open Real Interval:

$\left[{a \,.\,.\, b}\right]^\circ = \left({a \,.\,.\, b}\right)$

where $\left[{a \,.\,.\, b}\right]^\circ$ denotes the interior of $\left[{a \,.\,.\, b}\right]$.

Hence the result, by definition of regular open.

$\blacksquare$


Sources