Base (topology)

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.[1] 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 UγVα but also meets