1) B={(a,b)|a,b are rational numbers} is a countable base for the usual topology for the real numbers.

2) For any topologies T1 and T2 for X, if B1 is a base for T1 and B1 is a subset of T1, then T1 is a subset of T2

Def1: Bp is a local base for p in (X,T) iff p is an element of B and B is an element of T for every B an element of Bp; and when p is an ellement of G is an element of T, then there is Bg and element of Bp such that p is an element of Bg, which is a subset of G.

Def2: B is a base for the topology T iff B= the union of all Bp st p is an element of X, where each Bp is a loca base for p in the topological space (X,T).