Rationals are Everywhere Dense in Reals

Theorem

Topology

Let $\struct {\R, \tau_d}$ denote the real number line with the usual (Euclidean) topology.

Let $\Q$ be the set of rational numbers.


Then $\Q$ is everywhere dense in $\struct {\R, \tau_d}$.


Normed Vector Space

Let $\struct {\R, \size {\, \cdot \,}}$ be the normed vector space of real numbers.

Let $\Q$ be the set of rational numbers.


Then $\Q$ are everywhere dense in $\struct {\R, \size {\, \cdot \,}}$


Sources


This article is complete as far as it goes, but it could do with expansion.
In particular: Include the elementary proof given in this source
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  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): dense set
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