Closed Real Interval is Closed Set

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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 closed interval of $\R$.


Then $\left[{a \,.\,.\, b}\right]$ is a closed set of $\R$.


Proof

\(\displaystyle \left[{a \,.\,.\, b}\right]\) \(=\) \(\displaystyle \left\{ {x \in \R: x \ge a \land x \le b}\right\}\) Definition of Closed Real Interval
\(\displaystyle \implies \ \ \) \(\displaystyle \R \setminus \left[{a \,.\,.\, b}\right]\) \(=\) \(\displaystyle \R \setminus \left\{ {x \in \R: x \ge a \land x \le b}\right\}\)
\(\displaystyle \) \(=\) \(\displaystyle \left\{ {x \in \R: x < a \lor x > b}\right\}\) De Morgan's Laws: Disjunction of Negations
\(\displaystyle \) \(=\) \(\displaystyle \left({-\infty \,.\,.\, a}\right) \cup \left({b \,.\,.\, \infty}\right)\) Definition of Open Real Interval

From the corollary to Open Real Interval is Open Set, both $\left({-\infty \,.\,.\, a}\right)$ and $\left({b \,.\,.\, \infty}\right)$ are open sets in $M$.

From Union of Open Sets of Metric Space is Open it follows that $\left({-\infty \,.\,.\, a}\right) \cup \left({b \,.\,.\, \infty}\right)$ is open in $\R$.

But $\left({-\infty \,.\,.\, a}\right) \cup \left({b \,.\,.\, \infty}\right)$ is the relative complement of $\left[{a \,.\,.\, b}\right]$ in $\R$.

The result follows by definition of closed set.

$\blacksquare$


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