If B is a basis for the topology of X and C is a basis for the topology of Y, then the collection D = {B × C | B ∈ B and C ∈ C} is a basis for the topology of X ×Y. Subspaces. Consider R with the standard topology as well as R ‘: the real numbers with the lower limit topology, whose basis consists of the intervals [a,b). Base for a topology. It is a square in the plane C = R2 with some of the ‘boundary’ included and some not. It is again neither open 15. Homeomorphisms 16 10. The Uptime Institute Tier Standard: Topology is an objective basis for comparing the functionality, capacity, and expected availability (or performance) of a particular site infrastructure design topology against other sites, or for comparing a group of sites. The topology generated by is finer than (or, respectively, the one generated by ) iff every open set of (or, respectively, basis element of ) can be represented as the union of some elements of . careful, we should really say that we are using the standard absolute value metric on R and the corresponding metric topology — the usual topology to use for R.) An example that is perhaps more satisfying is fz= x+iy2C : 0 x;y<1g. Any such set can be decomposed as the union S a