Closure of Real Interval is Closed Real Interval
Theorem
Let $I$ be a non-empty real interval such that one of these holds:
- $I = \openint a b$
- $I = \hointr a b$
- $I = \hointl a b$
- $I = \closedint a b$
Let $I^-$ denote the closure of $I$.
Then $I^-$ is the closed real interval $\closedint a b$.
Proof 1
There are four cases to cover:
- $(1): \quad$ Let $I = \openint a b$.
From Closure of Open Real Interval is Closed Real Interval:
- $I^- = \closedint a b$
$\Box$
- $(2): \quad$ Let $I = \hointr a b$.
From Closure of Half-Open Real Interval is Closed Real Interval:
- $I^- = \closedint a b$
$\Box$
- $(3): \quad$ Let $I = \hointl a b$.
From Closure of Half-Open Real Interval is Closed Real Interval:
- $I^- = \closedint a b$
$\Box$
- $(4): \quad$ Let $I = \closedint a b$.
From Closed Real Interval is Closed in Real Number Line:
- $I$ is closed in $\R$.
From Set is Closed iff Equals Topological Closure:
- $I^- = \closedint a b$
$\Box$
Thus all cases are covered.
The result follows by Proof by Cases.
$\blacksquare$
Proof 2
Let $I$ be one of the intervals as specified in the exposition.
Note that:
- $(1): \quad$ By Condition for Point being in Closure, $x \in I^-$ if and only if every open set in $\R$ containing $x$ contains a point in $I$.
- $(2): \quad$ From Union of Open Sets of Metric Space is Open, every open set in $\R$ is a union of open intervals.
Thus we also have that $x \in I^-$ if and only if every open interval containing $x$ also contains a point in $I$.
This equivalence will be made use of throughout.
Lemma 1
- $x \in \closedint a b \implies x \in I^-$
$\Box$
Lemma 2
- $x \notin \closedint a b \implies x \notin I^-$
$\Box$
By the two lemmas proven above:
- $\closedint a b = I^-$
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.7$: Definitions: Examples $3.7.13$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): closure
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): closure