RebYidd23: Of course it would be assur. A guy working out on the treadmill could be led to have inappropriate thoughts about who might have been running on that treadmill 24 hours earlier. A slippery slope to pritzus and worse.
Obviously its not assur given that Agudah and other frum mosdos have events at advertised venues highlight the fact that they provide swimming pools with separate hours for men and women. A more complex but pressing shaylah for the armchair CR poskim is whether it is mutar for men and women to play on a golf course at the same time (in separate foursomes of course).
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