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initial topology
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(Definition)
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Let , be any family of topological spaces. We say that a topology
on is initial with respect to the family of mappings
, , if
is the coarsest topology on which makes all 's continuous.
The initial topology is characterized by the condition that a map
is continuous if and only if every
is continuous.
Sets
is open in 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 [2], weak topology [4] 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.)
- 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
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"initial topology" is owned by kompik.
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Cross-references: categorical, categories, structures, initial source, category theory, functional analysis, weak topology, generated by, embedding, subspace topology, product topology, base, intersections, finite, open, map, continuous, mappings, topological spaces
There are 3 references to this entry.
This is version 8 of initial topology, born on 2005-09-10, modified 2007-10-06.
Object id is 7368, canonical name is InitialTopology.
Accessed 2376 times total.
Classification:
| AMS MSC: | 54B99 (General topology :: Basic constructions :: Miscellaneous) |
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Pending Errata and Addenda
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