The ε-approximated complete invariance property

In the present paper we introduce a generalization of complete invariance property (CIP) for metric spaces, which will call ε-approximated (ε-ACIP). For our goals, use so called degree nondensifiability (DND) which, roughly speaking, measures (in specified sense) distance from bounded space to its class Peano continua. Our main result relates ε-ACIP with DND and, in particular, proves that densifiable has each ε>0. Also, some essentials differences between CIP and are shown.