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  "body": "\n## Definition\n\nLet $I$ be a set and let $(X_i)_{i \\in I}$ be a family of [[topological spaces|0CQHFPGJ45]].\nThe disjoint union of the family is the set\n$$X = \\bigsqcup_{i \\in I} X_i = \\set{(i, x) : i \\in I, \\ x \\in X_i}.$$\nFor each $i \\in I$, the canonical injection $\\iota_i \\colon X_i \\to X$ is $\\iota_i(x) = (i, x)$.\nThe *disjoint union topology* on $X$ is the set of all $U \\subseteq X$ such that $\\iota_i^{-1}(U)$ is [[open|02BH3MCBQB]] in $X_i$ for every $i \\in I$.\n\n## Remarks\n\nIt is the [[final topology|6ERJRRZK32]] on $X$ for the family of maps $\\iota_i$.\nEach $\\iota_i$ is an open [[topological embedding|00ZG9Y9TR4]], and the image of each $X_i$ is open and closed in $X$.\nUniversal property: a map $f \\colon X \\to Y$ to a topological space $Y$ is [[continuous|0VGFHTM75Y]] if and only if $f \\circ \\iota_i$ is continuous for every $i$.\n\nExample: if each $X_i$ is a one-point space, then every subset of $X$ is open, so $X$ is the discrete space on $I$.\nIndeed, the preimage of any subset under each $\\iota_i$ is empty or a point, and both are open.\n\nNon-example: let $X_0$ and $X_1$ be two copies of a two-point space with the trivial topology, and let $X = X_0 \\sqcup X_1$.\nThe trivial topology on the four-point set $X$ is not the disjoint union topology.\nThe set $\\iota_0(X_0)$ has open preimages in both pieces, but it is not open in the trivial topology.\n\nNotation: $\\bigsqcup_i X_i$ or $\\coprod_i X_i$; for two spaces, $X_1 \\sqcup X_2$.\n",
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