Anyone is a pronoun that indicates any person. It is used when there are no qualifications to the grouping. Something could belong to anyone if there are no distinguishing marks or unique factors.
Unlike anytime and anymore, anyone is not simply an adverb form of any one and they are not interchangeable outside of the United States.
Any one is a term that means any single object or person. It is generally not listed in dictionaries except perhaps to distinguish the differences with anyone.
Several phrases use the term anyone and slightly change the meaning for each.
Anyone who is anyone means people who are famous or respected in a social sphere.
To be anyone’s is to welcome flirtation (or more) from everyone without specific standards.
It could be anyone’s game if the teams are matched in skill and scoring. This phrase is used for all contests, whether physical or mental.
Anyone willing to part with between £20,000 and £200,000 for a week’s holiday can live in the homes of the rich and famous while they are away, order their staff around, and make use of their cars, tennis courts and swimming pools. [The Guardian]
Impressively, everything is available to tweak and tune if you so desire from the very start but the stock settings on each car are good enough to get you around any one of the current 110 track variations. [Scottish Daily Record]
In the United States, spiritual mobilization has long been anyone’s game. [The New York Times]