Coherent topology

(Redirected from Topological union)

In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a topological union of those subspaces. It is also sometimes called the weak topology generated by the family of subspaces, a notion that is quite different from the notion of a weak topology generated by a set of maps.[1]

Definition

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Let   be a topological space and let   be a family of subsets of   each with its induced subspace topology. (Typically   will be a cover of  .) Then   is said to be coherent with   (or determined by  )[2] if the topology of   is recovered as the one coming from the final topology coinduced by the inclusion maps   By definition, this is the finest topology on (the underlying set of)   for which the inclusion maps are continuous.   is coherent with   if either of the following two equivalent conditions holds:

  • A subset   is open in   if and only if   is open in   for each  
  • A subset   is closed in   if and only if   is closed in   for each  

Given a topological space   and any family of subspaces   there is a unique topology on (the underlying set of)   that is coherent with   This topology will, in general, be finer than the given topology on  

Examples

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Topological union

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Let   be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection   Assume further that   is closed in   for each   Then the topological union   is the set-theoretic union   endowed with the final topology coinduced by the inclusion maps  . The inclusion maps will then be topological embeddings and   will be coherent with the subspaces  

Conversely, if   is a topological space and is coherent with a family of subspaces   that cover   then   is homeomorphic to the topological union of the family  

One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings.

One can also describe the topological union by means of the disjoint union. Specifically, if   is a topological union of the family   then   is homeomorphic to the quotient of the disjoint union of the family   by the equivalence relation   for all  ; that is,  

If the spaces   are all disjoint then the topological union is just the disjoint union.

Assume now that the set A is directed, in a way compatible with inclusion:   whenever  . Then there is a unique map from   to   which is in fact a homeomorphism. Here   is the direct (inductive) limit (colimit) of   in the category Top.

Properties

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Let   be coherent with a family of subspaces   A function   from   to a topological space   is continuous if and only if the restrictions   are continuous for each   This universal property characterizes coherent topologies in the sense that a space   is coherent with   if and only if this property holds for all spaces   and all functions  

Let   be determined by a cover   Then

  • If   is a refinement of a cover   then   is determined by   In particular, if   is a subcover of     is determined by  
  • If   is a refinement of   and each   is determined by the family of all   contained in   then   is determined by  
  • Let   be an open or closed subspace of   or more generally a locally closed subset of   Then   is determined by  
  • Let   be a quotient map. Then   is determined by  

Let   be a surjective map and suppose   is determined by   For each   let  be the restriction of   to   Then

  • If   is continuous and each   is a quotient map, then   is a quotient map.
  •   is a closed map (resp. open map) if and only if each   is closed (resp. open).

Given a topological space   and a family of subspaces   there is a unique topology   on   that is coherent with   The topology   is finer than the original topology   and strictly finer if   was not coherent with   But the topologies   and   induce the same subspace topology on each of the   in the family   And the topology   is always coherent with  

As an example of this last construction, if   is the collection of all compact subspaces of a topological space   the resulting topology   defines the k-ification   of   The spaces   and   have the same compact sets, with the same induced subspace topologies on them. And the k-ification   is compactly generated.

See also

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Notes

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  1. ^ Willard, p. 69
  2. ^   is also said to have the weak topology generated by   This is a potentially confusing name since the adjectives weak and strong are used with opposite meanings by different authors. In modern usage the term weak topology is synonymous with initial topology and strong topology is synonymous with final topology. It is the final topology that is being discussed here.

References

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  • Tanaka, Yoshio (2004). "Quotient Spaces and Decompositions". In K.P. Hart; J. Nagata; J.E. Vaughan (eds.). Encyclopedia of General Topology. Amsterdam: Elsevier Science. pp. 43–46. ISBN 0-444-50355-2.
  • Willard, Stephen (1970). General Topology. Reading, Massachusetts: Addison-Wesley. ISBN 0-486-43479-6. (Dover edition).