Coherent Topology - Topological Union

Topological Union

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

together 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 {X_α}.

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

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

for all α, β in A. That is,

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