# Closure of Rational Interval is Closed Real Interval

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## Theorem

Let $\struct {\R, \tau_d}$ be the real numbers under the usual (Euclidean) topology.

Let $\struct {\Q, \tau_d}$ be the rational number space under the same topology.

Let $a, b \in \R$ such that $a < b$.

Let $\Bbb I \subseteq \R$ be an interval of $\R$

Then the closure of the set :

- $\Bbb I \cap \Q$

is the closed real interval $\closedint a b$.

## Proof

Let $\Bbb I$ be an open real interval.

From Closure of Real Interval is Closed Real Interval:

- $\Bbb I^- = \closedint a b$

From Closure of Rational Numbers is Real Numbers:

- $\Q^- = \R$

From Closure of Intersection is Subset of Intersection of Closures:

- $\paren{\Bbb I \cap \Q}^- \subseteq \Bbb I^- \cap \Q^-$

From Intersection with Subset is Subset:

- $\Bbb I^- \cap \Q^- = \closedint a b$

and so:

- $\paren {\Bbb I \cap \Q}^- \subseteq \closedint a b$

From Rationals are Everywhere Dense in Topological Space of Reals:

- $\closedint a b \subseteq \paren {\Bbb I \cap \Q}^-$

and the result follows.

$\blacksquare$