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initial topology∗
kompik†
2013-03-21 19:41:26
Let Xi , i ∈ I be any family of topological spaces. We say that a topology T
on X is initial with respect to the family of mappings fi : X → Xi , i ∈ I, if T
is the coarsest topology on X which makes all fi ’s continuous.
The initial topology is characterized by the condition that a map g : Y → X
is continuous if and only if every fi ◦ g : Y → Xi 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 embedding.
The initial topology is sometimes called topology generated by a family
of mappings [?], weak topology [?] 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.)
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 Initial topology
∗ hInitialTopologyi
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