Fixed points for mappings defined on pseudometric spaces

In this article, the distinct classes of continuous pseudometrics and metrics (perfect, quasi-perfect, sequentially complete pseudometrics and metrics) are defined and studied in depth. The conditions under which the set of fixed points of a given mapping of a space with concrete pseudometric is non-empty are determined. Some examples are proposed. For spaces with pseudometrics there are proved the Bishop-Phelps, Takahashi, Caristy and other theorems.