Basis for a topology?

[pivoxa15]

1. The problem statement, all variables and given/known data
Does a basis exist for all sets with a topology? In other words do all topological spaces with topology T contain a basis for T?

[AKG]

Of course. T itself is always a basis for T. In general, just as with vector spaces, there is no single unique basis for a given topology, but there is always a basis for it. The only time a topology might have a unique basis is if it's the indiscrete topology, or something like that.