Graphensätze für topologische Räume

In generalising a closed graph theorem of Dektjarev 5] we show that each almost closed and uniformly almost continuous relation of a quasi-uniform space to a hypercomplete quasi-uniform space is uniformly continuous. The hypothesis of being almost closed is necessary and actually weaker than the requirement that the relation considered has a closed graph. Since each topological space can be quasi-uniformized, one obtains closed graph theorems for topological spaces.As consequences we get an analogous closed graph theorem for locally uniformly almost continuous relations and a theorem concerning the completeness of the range of each continuous uniformly almost open mapping of a hypercomplete space to a uniform space.Finally we prove a closed graph theorem for relations to locally compact spaces without referring to any quasi-uniformization and thus without assumptions and statements concerning uniformity.