Having just dabbled in 7.1of rationality in extensive games by Andres Perea, I got an impression all constructions of universal type space are restricted to finite players. What's the obstacle to extend it to countably infinite players? For example, is it trivial to extend them to cover OLG games in which each stage game is only played by finite players?
Countably infinite is easy to extend to, no obstacles. You can use results by Moss and Viglizzo (2006). I will also shamelessly promote my own work Universal type structures with unawareness that mathematically treats each of (possibly countably infinitely many) awareness levels as a player.