Para-uniformities, para-proximities, and H-closed extensions (Q1819779)

The notion of a para-uniformity is a generalization of that of a uniformity, where essentially, the entourages are not required to cover the whole space but a dense subset. There is an analogous notion of para- proximity. It is shown that every topology is obtainable from an appropriate para-uniformity. Cauchy filters and completions are defined. Two natural types of extensions of a space are obtained in this manner, namely the simple and strict extensions. Certain families of para- uniformities on a space X are called superstructures. These superstructures give rise to a completion of X. It is shown that these characterize the strict H-closed extensions. Fedorchuk has previously proved a similar thing for the so-called R-equivalence classes of H- closed extensions of the space X.