# 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 $\left[{a \,.\,.\, b}\right) \subset \R$ be a half-open interval of $\R$.

Then $\left[{a \,.\,.\, b}\right)$ is neither an open set nor a closed set of $\R$.

Similarly, the half-open interval $\left({a \,.\,.\, b}\right] \subset \R$ is neither an open set nor a closed set of $\R$.

## Proof

From Half-Open Real Interval is not Open Set we have that $\left[{a \,.\,.\, b}\right)$ is not an open set of $\R$.

From Half-Open Real Interval is not Closed in Real Number Line we have that $\left[{a \,.\,.\, b}\right)$ is not a closed set of $\R$.

$\blacksquare$

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: Compactness - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $3.7$: Definitions: Examples $3.7.3 \ \text{(a)}$