initial topology

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 $f_{i}\colon X\to X_{i}$ , $i\in I$ , 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 $g\colon Y\to X$ is continuous if and only if every $f_{i}\circ g\colon Y\to X_{i}$ 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 (http://planetmath.org/GeneralizedCartesianProduct) 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 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.)

References

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 http://en.wikipedia.org/wiki/Initial_topologyInitial topology

Title	initial topology
Canonical name	InitialTopology
Date of creation	2013-03-22 15:30:26
Last modified on	2013-03-22 15:30:26
Owner	kompik (10588)
Last modified by	kompik (10588)
Numerical id	11
Author	kompik (10588)
Entry type	Definition
Classification	msc 54B99
Related topic	producttopology
Related topic	subspacetopology
Related topic	ProductTopology
Related topic	IdentificationTopology
Related topic	Coarser