Disjoint union (topology)

In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, two or more spaces may be considered together, each looking as it would alone.

The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.

Let {X_i : 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 ${\displaystyle \varphi _{i}(x)=(x,i)}$). The disjoint union topology on X is defined as the finest topology on X for which the canonical injections {φ_i} are continuous.

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 ${\displaystyle \varphi _{i}^{-1}(U)}$ is open in X_i for each i ∈ I.

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