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We characterize the pseudo-arc as well as P-adic pseudo-solenoids (for a setof primes P) as generic structures, arising from a natural game in which twoplayers alternate in building an inverse sequence of surjections. The secondplayer wins if the limit of this sequence is homeomorphic to a concrete (fixedin advance) space, called generic whenever the second player has a winningstrategy.  For this aim, we develop a new approximate Fra\"iss\'e theory, in order torealize the above-mentioned objects (the pseudo-arc and the pseudo-solenoids)as Fra\"iss\'e limits. Our framework extends the discrete Fra\"iss\'e theory,both classical and projective, and is also suitable for working directly withcontinuous maps on metrizable compacta.  We show, in particular, that, when playing with continuous surjectionsbetween non-degenerate Peano continua, the pseudo-arc is always generic. Theuniversal pseudo-solenoid appears to be generic over all surjections betweencircle-like continua.

Hereditarily indecomposable continua as generic mathematical structures