  We characterize the pseudo-arc as well as P-adic pseudo-solenoids (for a set
of primes P) as generic structures, arising from a natural game in which two
players alternate in building an inverse sequence of surjections. The second
player wins if the limit of this sequence is homeomorphic to a concrete (fixed
in advance) space, called generic whenever the second player has a winning
strategy.
  For this aim, we develop a new approximate Fra\"iss\'e theory, in order to
realize 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 with
continuous maps on metrizable compacta.
  We show, in particular, that, when playing with continuous surjections
between non-degenerate Peano continua, the pseudo-arc is always generic. The
universal pseudo-solenoid appears to be generic over all surjections between
circle-like continua.
