# Rationals are Everywhere Dense in Reals

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## 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

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**dense set**