In an abstract category with suitable notions of subobject, closure and point, we discuss the separation axioms and . Each of the arising subcategories is reflective. We give an iterative construction of the reflectors and present characteristic examples.
For every topological property , we define the class of -approximable spaces which consists of spaces X having a countable closed cover such that the “section” has the property for each . It is shown that every -approximable compact space has , if is one of the following properties: countable tightness, -scatteredness with respect to character, -closedness, sequentiality (the last holds under MA or ). Metrizable-approximable spaces are studied: every compact space in this class has...
In the present paper, we introduce and study the concept of -closed sets in biclosure spaces and investigate its behavior. We also introduce and study the concept of -continuous maps.
It is known that for a nonempty topological space and a nonsingleton complete lattice endowed with the Scott topology, the partially ordered set of all continuous functions from into is a continuous lattice if and only if both and the open set lattice are continuous lattices. This result extends to certain classes of -distributive lattices, where is a subset system replacing the system of all directed subsets (for which the -distributive complete lattices are just the continuous...
We continue the study of 1/2-homogeneity of the hyperspace suspension of continua. We prove that if X is a decomposable continuum and its hyperspace suspension is 1/2-homogeneous, then X must be continuum chainable. We also characterize 1/2-homogeneity of the hyperspace suspension for several classes of continua, including: continua containing a free arc, atriodic and decomposable continua, and decomposable irreducible continua about a finite set.