Poincare Project: A Basis for a Topology

Because of the requirement that unions and finite intersections of open sets must also be open sets, you don't need to specify every open set in order to define a topology. You can characterise a topology by describing a certain class of open sets from which the other open sets can be calculated.

Such a class is called a basis for the topology.

Because members of the basis are themselves open sets, once we have a basis we can generate all the other open sets by taking unions.

A random selection of subsets of X isn't always going to give as a basis for a topology on X anymore than it gives us a topology, so what restrictions exists on a basis the ensure it can generate a topology?

Clearly every element in the set X must appear in at least one basis open set. Otherwise that element would miss out on being in any open sets (and we know that, by definition, X itself must be open).

There is one more requirement, however, that must be met. Consider X = {a, b, c}. The open sets {a, b}, {b, c} cannot form a basis because if {a, b} and {b, c} are open then the intersection {b} must be. But {b} cannot be open because it isn't the union of basis open sets.

To avoid this, we have the additional requirement on a basis as follows:

if x is in the intersection of two basis open sets then x must also be in a third basis open set which is a subset of the intersection.

This, along with the requirement that every element must appear in at least one basis open set is sufficient to ensure that one has a basis for a topology.

UPDATE: next post

The original post was in the category: poincare_project but I'm still in the process of migrating categories over.