Relations between Semiuniform Convergence Spaces and Merotopic Spaces (including Nearness Spaces)

Abstract

M. Katětov [80] originally introduced filter spaces in the realm of his merotopic spaces (studied in the same paper) and called them filter-merotopic spaces. We start the present chapter with this alternative description of filter spaces which have been introduced in chapter 1 and which have also been described in the framework of semiuniform convergence spaces in chapter 2. In other words, a filter space may be regarded as a (filter-)merotopic space or as a Fil-determined semiuniform convergence space. Furthermore, the construct Fil is bicoreflectively embedded in the construct Mer of merotopic spaces, whereas it is bireflectively and bicoreflectively embedded into SUConv. As already mentioned in the introduction of this book, the formation of subspaces in Top (or Tops) is not satisfactory. The reason becomes clear, when subspaces of symmetric topological spaces are formed in SUConv: they are not topological in general unless they are closed. Since symmetric topological spaces may be regarded as complete filter spaces, subspaces of them, formed in SUConv, are filter spaces (regarded as semiuniform convergence spaces). Thus, in order to answer the question how subspaces (in SUConv) of symmetric topological spaces, called subtopological spaces, can be characterized axiomatically, we may focus our interest to Fil. Such an axiomatic characterization in terms of filters is given in the second part of this chapter. Another characterization due to H.L. Bentley [10] is found, when the description of filter spaces in the realm of merotopic spaces is used, namely a filter space is subtopological iff its corresponding merotopic space is a nearness space. Nearness spaces have been introduced and studied first by H. Herrlich [62].