Closed Real Interval is not Open Set

Theorem

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

Let $\closedint a b \subset \R$ be a closed interval of $\R$.


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


Proof

From Closed Real Interval is Neighborhood Except at Endpoints, $a$ and $b$ have no open $\epsilon$-ball lying entirely in $\closedint a b$.

The result follows by definition of open set.

$\blacksquare$


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): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.9$
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