Fine topologies as examples of non-Blumberg Baire spaces
In his paper "Continuous mappings on continua" [5], T. Maćkowiak collected results concerning mappings on metric continua. These results are theorems, counterexamples, and unsolved problems and are listed in a series of tables at the ends of chapters. It is the purpose of the present paper to provide solutions (three proofs and one example) to four of those problems.
Let be a zero-dimensional space and be the set of all continuous real valued functions on with countable image. In this article we denote by (resp., the set of all functions in with compact (resp., pseudocompact) support. First, we observe that (resp., ), where is the Banaschewski compactification of and is the -compactification of . This implies that for an -compact space , the intersection of all free maximal ideals in is equal to , i.e., . By applying methods of functionally...
A function mapping the topological space to the space is called a z-open function if for every cozeroset neighborhood of a zeroset in , the image is a neighborhood of in . We say has the z-separation property if whenever , are cozerosets and is a zeroset of such that , there is a zeroset of such that . A surjective function is z-open if and only if it maps cozerosets to cozerosets and has the z-separation property. We investigate z-open functions and other functions...