Let , be any family of topological spaces. We say that a topology on is initial with respect to the family of mappings , , if is the coarsest topology on which makes all 's continuous.

The initial topology is characterized by the condition that a map is continuous if and only if every is continuous.

Sets is open in form a subbase for the initial topology, their finite intersections form a base.

E.g. the product topology is initial with respect to the projections and a subspace topology is initial with respect to the embedding.

The initial topology is sometimes called topology generated by a family of mappings [2], weak topology [4] or projective topology. (The term weak topology is used mainly in functional analysis.)

From the viewpoint of category theory, the initial topology is an initial source. (Initial structures, which are a natural generalization of the initial topology, play an important rôle in topological categories and categorical topology.)

Bibliography

1

J. Adámek, H. Herrlich, and G. Strecker, Abstract and concrete categories, Wiley, New York, 1990.

2

R. Engelking, General topology, PWN, Warsaw, 1977.

3

M. Hušek, Categorical topology, Encyclopedia of General Topology (K. P. Hart, J.-I. Nagata, and J. E. Vaughan, eds.), Elsevier, 2003, pp. 70-71.

4

S. Willard, General topology, Addison-Wesley, Massachussets, 1970.

5

Wikipedia's entry on Initial topology