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A space      X X       is said to have the complete invariance property (CIP) if every nonempty closed subset of      X X       is the fixed point set of some self-map of      X X      . An example is given to show that there is an      L C ∞  L{C^\infty }      ,      C ∞  {C^\infty }       continuum      X X       which does not have CIP. Moreover,      X X       is a wedge of two      L C ∞  L{C^\infty }      ,      C ∞  {C^\infty }       continua each of which has CIP.

Fixed point sets of 𝐿𝐶^{∞}, 𝐶^{∞} continua