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  "body": "\n## Definition\n\nLet $Y$ be a set, let $I$ be a set, and for each $i \\in I$ let $X_i$ be a [[topological space|0CQHFPGJ45]] and $f_i \\colon X_i \\to Y$ a map.\nThe *final topology* on $Y$ for the family $(f_i)_{i \\in I}$ is the set of all $U \\subseteq Y$ with the following property.\nFor every $i \\in I$, the [[preimage|01AHG4G9RT]] $f_i^{-1}(U)$ is [[open|02BH3MCBQB]] in $X_i$.\n\n## Remarks\n\nThis family of sets is a topology, because preimages commute with unions and intersections.\nIt is the [[finest|072J68M6HX]] topology on $Y$ for which every $f_i$ is [[continuous|0VGFHTM75Y]].\nUniversal property: for a topological space $Z$, a map $g \\colon Y \\to Z$ is continuous for the final topology if and only if $g \\circ f_i$ is continuous for every $i$.\n\nExample: for a surjection $q \\colon X \\to Y$ from a space $X$, the final topology for $q$ is the [[quotient topology|0NQ1H2X495]].\nExample: for the injections of the pieces into a disjoint union, the final topology is the [[disjoint union topology|ZXJKSB30WF]].\n\nNon-example: let $q \\colon \\R \\to \\R$ be the identity, with the usual topology on the source.\nThe trivial topology $\\set{\\varnothing, \\R}$ on the target makes $q$ continuous but is not the final topology.\nThe set $(0, 1)$ has an open preimage, yet it is not open in the trivial topology.\n\nThe final topology is also called the strong topology or the coinduced topology for the family.\n",
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