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