Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group

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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