B-11 - Rim(P)-Spaces

Abstract

All topological spaces under consideration are assumed to be Hausdorff and all maps are assumed to be continuous and onto in this chapter. If P is a topological property and X a topological space, then X is said to be rim-P if X admits a base of open sets with boundaries having the property P. Some of the spaces that are studied with natural rim-properties are rim-finite spaces, rim-countable spaces, rim-metrizable spaces, rim-scattered spaces, and rim-compact spaces. If X is a 0-dimensional space, then it has a base of open sets with empty boundaries. Hence, such spaces are rim-P for any property P. Therefore, it is interesting to study rim-properties of spaces that are not 0-dimensional.The term locally peripherally P is sometimes used instead of rim-P. The study of (metric) continua that are close to arcs, continua with various rim-properties, and the classification of curves are important research areas in Topology. Another study concerns those compact Hausdorff spaces and continua that are continuous images of compact ordered spaces. This chapter surveys the results on rim-finite, rim-countable, rim-metrizable, rim-scattered spaces—mainly in the realm of compact spaces—and also rim-compact and rim-separable spaces.