Continua which admit only certain classes of onto mappingsContinua which admit only certain classes of onto mappings Peer-Reviewed Journal Publication Δημοσίευση σε Περιοδικό με Κριτές 2015-12-011978enThe purpose of this article is to present a rather com plete study of those classes of continua which admit only confluent (resp. semi-confluent, weakly confluent, pseudo-confluent) onto mappings. The first results were obtained by H. Cook [3] who proved that if X is a hereditarily inde composable continuum, then every mapping from any continuum onto X is confluent, and by D. R. Read [20] who proved that the converse is true, that is, if X is a continuum such that every mapping from any continuum onto X is confluent, then X is hereditarily indecomposable. In what follows we study the class of continua X with the property that every mapping from any continuum onto X is weakly confluent. Finally, at the end of the paper we study the classes of continua X with the property that every mapping from any continuum onto X is semi-confluent (resp., pseudo-confluent>. 1. Definitions and Preliminaries By a continuum is meant a connected, compact, metric space. By a mapping is always meant a continuous function. A mapping f: X ~ Y of a continuum X onto a continuum Y is said to be confluent [2], semi-confluent [18], or weakly lThe first author was supported by a University of Saskatchewan postdoctoral fellowship.http://creativecommons.org/licenses/by/4.0/Topology Proceedings3347–362 Gryspolakis Ioakeim Γρυσπολακης Ιωακειμ Tymchatyn E.D. Topology