Is every pseudo-orbit of some homeomorphism near an exact orbit of a nearby homeomorphism?

We propose a relaxed notion of shadowing. In particular, for a homeomorphism f on a metric space, we ask whether every approximate orbit is near some actual orbit of some nearby system. We distinguish the cases where the nearby homeomorphism is close or arbitrarily close to f. We prove the relations between these notions and ordinary shadowing and present various examples. We finally discuss an application, a C-0-closing lemma for chain recurrent points of a homeomorphism on a topological manifold, not necessarily compact. This result leads to a characterization of explosion phenomena for various recurrent sets on such manifolds.