This is Kelly’s small deviation from the conventional definitions given for topological spaces and their bases. Notions arise in a different order.

A topology is a set T of sets such that

If T is a topology and X = ∪T then T is a topology for the topological space X in the conventional sense. ∅∊T because ∅ ⊆ T and ∪∅ = ∅. X∊T because T ⊆ T and ∪T = X. All of the conventional axioms are thus true here.

A base family is a set B of sets such that
if b ∊ B and c ∊ B and x ∊ b∩c then ∃d (x ∊ d & d ⊆ b∩c & d ∊ B).

If B is a base family then T = {∪b | b⊆B} is a topology that is a conventional topology for ∪B and B is a conventional base for T.

If S is a set of sets and B is the smallest set of sets closed under intersection that includes S, then B is a base family and S is a conventional sub-base for ∪S.

Any set S of sets is a sub-base. It is a conventional sub-base of ∪S. The set of finite intersections of members of S is a base.

Rationale for these Notions

It is conventional to start with the space and then say what it takes to be a topology for that space. Instead here we speak of what it takes to be a topology and only then reveal what space it is a topology of. This makes a short set of axioms even shorter and perhaps more memorable.

Topology first, topological space second.

Subsequent development is largely unaffected by these quibbles; there is no new math here.

Historical Topology

While there are many interesting small topologies, the case that historically motivated these notions was the real line. A large portion of topological results apply usefully to the real line. The base family consisting of the set of open intervals produces the conventional topology for the real line. B = {{z | x < z & z < y} | x∊R & y ∊R & x < y} is a base for the conventional topology for R. For the real line the topology is the set of denumerable unions of members of B.

If R above is any totally ordered set, instead of the reals, then B is this base family for the order topology for T.

If T and U are base families then T⊗U = {t×u | t∊T & v∊V} is a base family for the topological space S = ∪T×∪U. It is called the product topology for S. If R' is the topology for the reals, then the product topology for R'⊗R' is the conventional topology for the Euclidean plane. Another base family that produces the same topology for the plane is the set of discs of arbitrary radii about arbitrary points in the plane. This is thus the topology that you get via the metric on the plane. All this works in n dimensions as well.