# Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group

## Theorem

Let $\R_{>0}$ be the set of strictly positive real numbers, i.e. $\R_{>0} = \left\{{ x \in \R: x > 0}\right\}$.

The structure $\left({\R_{>0}, \times}\right)$ is an uncountable abelian group.

## Proof

From Strictly Positive Real Numbers under Multiplication form Subgroup of Non-Zero Real Numbers we have that $\left({\R_{>0}, \times}\right)$ is a subgroup of $\left({\R_{\ne 0}, \times}\right)$, where $\R_{\ne 0}$ is the set of real numbers without zero, i.e. $\R_{\ne 0} = \R \setminus \left\{{0}\right\}$.

From Subgroup of Abelian Group is Abelian it also follows that $\left({\R_{>0}, \times}\right)$ is an abelian group.

Its infinite nature follows from the nature of real numbers.

$\blacksquare$

## Sources

- 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.06$