e-13 Generalized Metric Spaces III: Linearly Stratifiable Spaces and Analogous Classes of Spaces

This article is concerned with generalizations of concepts like stratifiability and metrizability to arbitrary infinite cardinalities, in a way that uses linear orders in key places. This has resulted in theories which are remarkably faithful generalizations of the theories of stratifiable, metrizable, etc. spaces. For metrizable spaces, the generalization is to the class of (Tychonoff ) spaces admitting separated uniformities with totally ordered bases; this class is usually referred to as the class of ωμ-metrizable spaces of arbitrary cardinality ωμ, but the term “linearly uniformizable spaces” will be mostly used here, under the convention that “spaces” refers to Hausdorff spaces. The class of linearly stratifiable spaces is a simultaneous generalization of linearly uniformizable spaces and of stratifiable spaces, and most of the theory of stratifiable spaces carries over, including the basic covering and separation properties of paracompactness and monotone normality. There are generalizations, along the same lines, of σ -spaces and semistratifiable spaces, as well as classes in between the linearly uniformizable spaces and linearly stratifiable spaces, generalizing M1 spaces and Nagata spaces. Other generalizations, such as the one of quasimetrizable spaces (quasi-metrics are defined like metrics but without symmetry of the distance function), are less well developed in the literature, and will only be touched on here. The usual definition of linear stratifiability is based on the definition of stratifiable spaces that says they are monotonically perfectly normal, so to speak; this definition is the case ωμ = ω of the definition of ωμ-stratifiable spaces, where ωμ is an infinite cardinal number. A space (X, τ) is said to be stratifiable over ωμ if it is a T1 space for which there is a map S :ωμ × τ → τ , called an ωμ-stratification which satisfies the following conditions.