Closed Real Interval is Closed Set
Jump to navigation
Jump to search
Theorem
Let $\R$ be the real number line considered as an Euclidean space.
Let $\closedint a b \subset \R$ be a closed interval of $\R$.
Then $\closedint a b$ is a closed set of $\R$.
Proof
\(\ds \closedint a b\) | \(=\) | \(\ds \set {x \in \R: x \ge a \land x \le b}\) | Definition of Closed Real Interval | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \R \setminus \closedint a b\) | \(=\) | \(\ds \R \setminus \set {x \in \R: x \ge a \land x \le b}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \set {x \in \R: x < a \lor x > b}\) | De Morgan's Laws: Disjunction of Negations | |||||||||||
\(\ds \) | \(=\) | \(\ds \openint {-\infty} a \cup \openint b \infty\) | Definition of Open Real Interval |
From the corollary to Open Real Interval is Open Set, both $\openint {-\infty} a$ and $\openint b \infty$ are open sets in $M$.
From Union of Open Sets of Metric Space is Open it follows that $\openint {-\infty} a \cup \openint b \infty$ is open in $\R$.
But $\openint {-\infty} a \cup \openint b \infty$ is the relative complement of $\closedint a b$ in $\R$.
The result follows by definition of closed set.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: Compactness
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Examples $3.7.2$
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.3$: Normed and Banach spaces. Topology of normed spaces