Open Real Interval is Open Set/Corollary

From ProofWiki
Jump to navigation Jump to search

Corollary to Open Real Interval is Open Set

Let $\R$ be the real number line considered as an Euclidean space.

Let $A := \left({a \,.\,.\, \infty}\right) \subset \R$ be an open interval of $\R$.

Let $B := \left({-\infty \,.\,.\, b}\right) \subset \R$ be an open interval of $\R$.


Then both $A$ and $B$ are open sets of $\R$.


Proof

From Open Real Interval is Open Set we have that for any $c \in \left({a \,.\,.\, b}\right)$ there exists an open $\epsilon$-ball of $c$ lying wholly within $\left({a \,.\,.\, b}\right)$.

When either of $a \to -\infty$ or $b \to \infty$ the result still holds.

The result follows by definition of open set.

$\blacksquare$