Half-Open Real Interval is not Open Set

Theorem

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

Let $\hointr a b \subset \R$ be a half-open interval of $\R$.


Then $\hointr a b$ is not an open set of $\R$.


Similarly, the half-open interval $\hointl a b \subset \R$ is not an open set of $\R$.


Proof

Let $\epsilon \in \R_{>0}$.

Let $\map {B_\epsilon} a$ be the open $\epsilon$-ball of $a$.

We have that $a - \epsilon < a$ and so $\map {B_\epsilon} a = \openint {a - \epsilon} {a + \epsilon}$ does not lie entirely in $\hointr a b$.

Thus $\hointr a b$ is not a neighborhood $a$.

It follows that $\hointr a b$ is not an open set of $\R$.


the argument also shows, mutatis mutandis, that $\hointl a b \subset \R$ is not an open set of $\R$.

$\blacksquare$


Sources

  • 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.9$
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