= Topological disjoint union
{title2=$\coprod_{i\in I}X_i$}
= Disjoint union topology
{synonym}
The topological disjoint union of <topological spaces> $X_i$ has the labelled <disjoint union> as its underlying <set>. A <subset> is <open> exactly when its <intersection> with each labelled copy $X_i$ is <open> in that copy. Each copy is a <clopen set>, and a <function> from the topological disjoint union is <continuous> exactly when its <restriction> to every copy is <continuous>. This follows immediately by testing <preimages> of <open sets> on each copy. A <topological space> is <homeomorphic> to the topological disjoint union of its <connected components> exactly when all those <connected components> are <open>.
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