# How can you construct a topology from a fundamental system of neighborhoods ?

• Oct 1st 2011, 04:39 AM
Aki
How can you construct a topology from a fundamental system of neighborhoods ?
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 $\displaystyle (K,\epsilon)$ consisting of a compact subset $\displaystyle K \subset D$ and a number $\displaystyle \epsilon > 0$, we consider the subset $\displaystyle V(K,\epsilon)$ of C(D) defined by

$\displaystyle f \in V(K,\epsilon) \Leftrightarrow |f(x)|\leq \epsilon, \; x \in K.$

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 $\displaystyle V(K,\epsilon)$ 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.