Closure of Rational Numbers is Real Numbers
Jump to navigation
Jump to search
Theorem
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Let $\struct {\Q, \tau_d}$ be the rational number space under the same topology.
Then:
- $\Q^- = \R$
where $\Q^-$ denotes the closure of $\Q$.
Proof
From Rationals are Everywhere Dense in Topological Space of Reals, $\Q$ is everywhere dense in $\R$.
It follows by definition of everywhere dense that $\Q^- = \R$.
$\blacksquare$