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Coherent topology

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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).