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 which is quite different from the notion of a weak topology generated by a set of maps.[1]

Definition[edit]

Let X be a topological space and let C = {Cα : α ∈ A} be a family of subspaces of X (typically C will be a cover of X). Then X is said to be coherent with C (or determined by C)[2] if X has the final topology coinduced by the inclusion maps

i_\alpha : C_\alpha \to X\qquad \alpha \in A.

By definition, this is the finest topology on (the underlying set of) X for which the inclusion maps are continuous.

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

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

Examples[edit]

Topological union[edit]

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

X^{set} = \bigcup_{\alpha\in A}X_\alpha

endowed with the final topology coinduced by the inclusion maps i_\alpha : X_\alpha \to X^{set}. 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

(x,\alpha) \sim (y,\beta) \Leftrightarrow x = y

for all α, β in A. That is,

X  \cong \coprod_{\alpha\in A}X_\alpha / \sim .

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: \alpha\le\beta whenever X_\alpha\subset X_\beta. Then there is a unique map from \varinjlim X_\alpha to X, which is in fact a homeomorphism. Here \varinjlim X_\alpha is the direct (inductive) limit (colimit) of {Xα} in the category Top.

Properties[edit]

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

f|_{C_\alpha} : C_\alpha \to Y\,

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 : XY.

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 {YCα}.

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

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

f_\alpha : f^{-1}(D_\alpha) \to D_\alpha\,

be the restriction of f to f−1(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[edit]

  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[edit]

  • Tanaka, Yoshio (2004). "Quotient Spaces and Decompositions". In K.P. Hart, J. Nagata, and J.E. Vaughan. 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).