This is a small deviation from the conventional definitions given for topologies and their bases.
I propose some notions in a different order.
An open family is a set T of sets such that
- if x∈T and y∈T then x∩y ∈T
- if t ⊆ T then ∪t ∈T
If T is an open family 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.
A base family is a set B of sets such that
- if b ∈ B then ∃d d ∈ b
- 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 {∪b | b⊆B} is an open family that is a topology for ∪B.
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.
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.
We could use “topology” in place of “open family”.
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 open family 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 family then {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.
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.