# 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 InitialTopology 2013-03-22 15:30:26 2013-03-22 15:30:26 kompik (10588) kompik (10588) 11 kompik (10588) Definition msc 54B99 producttopology subspacetopology ProductTopology IdentificationTopology Coarser