# Closure of Open Real Interval is Closed Real Interval

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## Theorem

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.

Let $\openint a b$ be an open interval of $\R$.

Then the closure of $\openint a b$ is the closed interval $\closedint a b$.

## Proof

From Limit Points of Open Real Interval, the limit points of $\openint a b$ consist of:

- the points $\openint a b$ itself

and

- the points $a$ and $b$.

By definition, the closure of $\openint a b$ is the union of $\openint a b$ and its limit points.

Hence the result, by definition of the closed interval $\closedint a b$.

$\blacksquare$