Half-Open Real Interval is neither Open nor Closed
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Theorem
Let $\R$ be the real number line considered as an Euclidean space.
Let $\hointr a b \subset \R$ be a right half-open interval of $\R$.
Then $\hointr a b$ is neither an open set nor a closed set of $\R$.
Similarly, the left half-open interval $\hointl a b \subset \R$ is neither an open set nor a closed set of $\R$.
Proof
From Half-Open Real Interval is not Open Set we have that neither $\hointr a b$ nor $\hointl a b$ is an open set of $\R$.
From Half-Open Real Interval is not Closed in Real Number Line we have that neither $\hointr a b$ nor $\hointl a b$ is a closed set of $\R$.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Examples $3.7.3 \ \text{(a)}$