initial topology

Let Xi, iI be any family of topological spacesMathworldPlanetmath. We say that a topology 𝒯 on X is initial with respect to the family of mappings fi:XXi, iI, if 𝒯 is the coarsest topology on X which makes all fi’s continuousPlanetmathPlanetmath.

The initial topology is characterized by the condition that a map g:YX is continuous if and only if every fig:YXi is continuous.

Sets 𝒮={fi-1(U):U is open in Xi} 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 embeddingMathworldPlanetmathPlanetmath.

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 theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath, the initial topology is an initial source. (Initial structuresMathworldPlanetmath, which are a natural generalizationPlanetmathPlanetmath of the initial topology, play an important rôle in topological categories and categorical topology.)


  • 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 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 CoarserPlanetmathPlanetmath