TITLE: Which topological spaces are (topological) groups?
QUESTION [16 upvotes]: General literature does not seem to offer a characterisation of topological groups among all topological spaces. Of course, being completely regular (uniform) is necessary, but separation properties, or indeed any sort of "niceness" like pseudo-metrisability are not sufficient, since topological groups, for instance, cannot have fixed point property.

REPLY [3 votes]: A necessary condition for a Hausdorff compact space to admit the structure of a topological group is the Suslin condition (I hope I am using proper terminology)
every family of pair-wise disjoint open sets is countable.

This is so because Hausdorff compact topological groups admit Haar measure.