Coherent topology

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

Let X be a topological space and let C = {Cα : α A} be a family of subsets of X with subspace topology. (Typically C will be a cover of X.) Then X is said to be coherent with C (or determined by C)[2] if the topology of X 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) X for which the inclusion maps are continuous. If C is a cover of X, then X is coherent with C if either of the following two equivalent conditions holds:

  • A subset U is open in X if and only if U Cα is open in Cα for each α A.
  • A subset U is closed in X if and only if U Cα is closed in Cα for each α A.

The above is not true if C does not cover X.

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

Examples

Topological union

Let be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection Xα Xβ. Assume further that Xα Xβ is closed in Xα for each α,β. Then the topological union X is the set-theoretic union

endowed with the final topology coinduced by the inclusion maps . The inclusion maps will then be topological embeddings and X will be coherent with the subspaces {Xα}.

Conversely, if X is coherent with a family of subspaces {Cα} that cover X, then X is homeomorphic to the topological union of the family {Cα}.

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 X is a topological union of the family {Xα}, then X is homeomorphic to the quotient of the disjoint union of the family {Xα} by the equivalence relation

for all α, β in A. That is,

If the spaces {Xα} 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 X, which is in fact a homeomorphism. Here is the direct (inductive) limit (colimit) of {Xα} in the category Top.

Properties

Let X be coherent with a family of subspaces {Cα}. A map f : X Y is continuous if and only if the restrictions

are continuous for each α A. This universal property characterizes coherent topologies in the sense that a space X is coherent with C if and only if this property holds for all spaces Y and all functions f : X Y.

Let X be determined by a cover C = {Cα}. Then

  • If C is a refinement of a cover D, then X is determined by D.
  • If D is a refinement of C and each Cα is determined by the family of all Dβ contained in Cα then X is determined by D.

Let X be determined by {Cα} and let Y be an open or closed subspace of X. Then Y is determined by {Y Cα}.

Let X be determined by {Cα} and let f : X Y be a quotient map. Then Y is determined by {f(Cα)}.

Let f : X Y be a surjective map and suppose Y is determined by {Dα : α A}. For each α A let

be the restriction of f to f1(Dα). Then

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

Notes

  1. Willard, p. 69
  2. X is also said to have the weak topology generated by C. 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

  • 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).
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