Let $X_i$ , $i\in I$ be any family of topological spaces. We say that a topology $\mathcal T$ on $X$ is initial with respect to the family of mappings $\Map {f_i}X{X_i}$ , $i\in I$ , if $\mathcal T$ is the coarsest topology on $X$ which makes all $f_i$ 's continuous.

The initial topology is characterized by the condition that a map $\Map gYX$ is continuous if and only if every $\Map {f_i \circ g}Y{X_i}$ is continuous.

Sets $\mathcal S=\{f_i^{-1}(U): U$ is open in $X_i\}$ 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