Hybrid topologies on the real line

Abstract

[EN] Given A ⊆ R, the Hattori space H(A) is the topological space (R, τA) where each a ∈ A has a τA-neighborhood base {(a−ε, a+ε) : ε > 0} and each b ∈ R − A has a τA-neighborhood base {[b, b + ε) : ε > 0}. Thus, τA may be viewed as a hybrid of the Euclidean topology and the lowerlimit topology. We investigate properties of Hattori spaces as well as other hybrid topologies on R using various combinations of the discrete, left-ray, lower-limit, upper-limit, and Euclidean topologies. Since each of these topologies is generated by a quasi-metric on R, we investigate hybrid quasi-metrics which generate these hybrid topologies.