Definition
Let {Xi : i ∈ I} be a family of topological spaces indexed by I. Let
be the disjoint union of the underlying sets. For each i in I, let
be the canonical injection (defined by ). The disjoint union topology on X is defined as the finest topology on X for which the canonical injections are continuous (i.e. the final topology for the family of functions {φi}).
Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in X if and only if its preimage is open in Xi for each i ∈ I.
Yet another formulation is that a subset V of X is open relative to X iff its intersection with Xi is open relative to Xi for each i.
Read more about this topic: Disjoint Union (topology)
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