  An approach to find a weak form of shadowing is developed. We consider
homeomorphisms of a compact metric space. It is proved that every
pseudotrajectory with sufficiently small errors contains at least one
subsequence that can be shadowed by a subsequence of an exact trajectory with
same indices. We study systems with so-called multishadowing property that is
any pseudotrajectory can be shadowed by a finite number of exact orbits.
Criteria for existence of $\varepsilon$-- networks whose iterations are
$\varepsilon$ -- networks are given. Relations between multishadowing and some
ergodic and topological properties of dynamical systems are discussed.
