Bases are ubiquitous throughout topology. The sets in a base for a topology, which are called basic open sets, are often easier to describe and use than arbitrary open sets. Many important topological definitions such as continuity and convergence can be checked using only basic open sets instead of arbitrary open sets. Some topologies have a base of open sets with specific useful properties that may make checking such topological definitions easier.
The Zariski topology on the spectrum of a ring has a base consisting of open sets that have specific useful properties. For the usual base for this topology, every finite intersection of basic open sets is a basic open set.
We shall work with notions established in (Engelking 1977, p. 12, pp. 127-128).
The point of computing the character and weight is to be able to tell what sort of bases and local bases can exist. We have the following facts:
Using the above notation, suppose that w(X) ≤ κ some infinite cardinal. Then there does not exist a strictly increasing sequence of open sets (equivalently strictly decreasing sequence of closed sets) of length ≥ κ+.
This map is injective, otherwise there would be α < β with f(α) = f(β) = γ, which would further imply Uγ ⊆ Vα but also meets