Certain Completeness Properties and Mappings

We prove, among the others, that the directed completeness is preserved under perfect mappings. We introduce some new notions such as nearly directed completeness, $$C$$ -directed completeness and directed completeness of a family. We prove that nearly directed completeness is preserved under continuous, open images and preimages with compact fibers, and consequently, we obtain that a topological group $$G$$ is nearly directed complete if and only if the quotient to a compact subgroup $$H$$ of $$G$$ of $$G$$ is nearly directed complete. We also give an upper bound for domains which improves given one in [7] and this is an answer to Question 11.10 in [3].