How can you construct a topology from a fundamental system of neighborhoods ?
In "Elementary Theory of Analytic Functions of One or Several Complex Variables" by Henri Cartan, it seems that a topology is uniquely determined in C(D), the vector space of continuous complex-valued functions in the open set D, by a fundamental system of neighborhoods.
The fundamental system of neighborhoods of o is defined as follows:
For any pair consisting of a compact subset and a number , we consider the subset of C(D) defined by
The neighborhoods of a point f are defined by translating the neighborhoods of o by f.
Then, Proposition 3.I. follows
Proposition 3.I.
C(D) has indeed a topology (invariant under translation) in which the sets form a fundamental system of neighborhoods of o. This topology is unique and can be defined by a distance which is invariant under translation.
Proof.
The uniqueness of the topology is obvious, because we know a fundamental system of neighborhoods of o, and ...
I know that a topology can be constructed by specifying all neighborhoods of each point x (for example Bourbaki "Elements of Mathematics: General Topology I.1.2 Proposition 2"), but I cannot understand how a topology is defined from a fundamental system of neighborhoods.