Character

Definition
The character of a topological space at a point is defined as the minimum over cardinalities for every basis at that point. The character of a topological space (overall) is defined as the supremum over all points, of the character at that point.

(If this value turns out to be finite, we take the character as countable).

Relation with other cardinal functions
The character is related to the weight. The weight is the minimum over cardinalities over every basis (for the whole space). The character is clearly bounded from above by the weight.

Particular cases
A topological space with countable character is termed a first-countable space.