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 ${f}_{i}:X\to {X}_{i}$, $i\in I$, 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 $g:Y\to X$ is continuous if and only if every ${f}_{i}\circ g:Y\to {X}_{i}$ 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^{} |