Sign of a Permutation

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, let $n\in\mathbb{N}$, with $[n]$ as in Intervals of Natural Numbers (ii), $S_{n}$ as in Permutations of a Set and the Set of Permutations of [n] (ii), the set $\mathrm{Inv}(\sigma)$ of inversions of $\sigma\in S_{n}$ as in Inversions of a Permutation of [n] (i), finite by Inversions of a Permutation Form a Finite Set, and the Sign Map Exists (i), the number of elements $\#A$ of a finite set $A$, and powers in $\mathbb{R}$ as in Commutative Rings, Fields and Ordered Fields: Standard Notation (iii).

(i) The sign $\operatorname{sgn}:S_{n}\to\mathbb{R}$ is the map $s$ of Inversions of a Permutation Form a Finite Set, and the Sign Map Exists (ii), so that $\operatorname{sgn}(\sigma)=(-1)^{\#\mathrm{Inv}(\sigma)}$ for every $\sigma\in S_{n}$.