Boards cannot be 'uncountably infinite' in size, right? At least not with conventional topology of square cells. That is per definition a mapping on Z x Z (where Z is the set of integers). But I think that even with countably infinite boards it is always possible to put a countably infinite number of pieces in between, with infinite distances behind those. Just keep putting pieces halfway in between all the pieces you have placed so far. No matter how often you divide infinity by 2, it remains infinite, so you can do that forever.
Playable, balanced variants for infinite boards?
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I was thinking, are there any chess variants that are actually worth playing where the board is unlimited in size? With moves themselves being finite but arbitrarilly large. Surely there has to be interesting variants with such a simple and canonical board?
Bonus question: what about boards where the pieces start infinitely far away from each other and the board is an uncountable infinity in size? I.e. where you could actually move an infinite distance and still have an infinite distance between you and another piece that moved an infinite distance in the same direction, with an infinite number of pieces being able to fit between them infinitely far apart? I'm almost certain this does not exist.