> separate women and men sports

Most professional sports I know of (in the US) technically don't have separate men leagues. There are women only leagues, and leagues open to anyone. There is no rule preventing women from playing in the NFL, NHL or MLB. But in practice if you don't enforce a minimum % of women players like I've seen in some amateur mixed leagues, it makes them men only.

The Olympics does split into men-only and women-only, I think.