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  "body": "\n## Definition\n\nLet $X$ be a set, let $I$ be a set, and for each $i \\in I$ let $Y_i$ be a [[topological space|0CQHFPGJ45]] and $f_i \\colon X \\to Y_i$ a map.\nThe *initial topology* on $X$ for the family $(f_i)_{i \\in I}$ is the [[coarsest|072J68M6HX]] topology on $X$ for which every $f_i$ is [[continuous|0VGFHTM75Y]].\nEquivalently, it is the topology generated by the [[subbasis|09S8DV34Y0]]\n$$\\set{f_i^{-1}(V) : i \\in I, \\ V \\text{ open in } Y_i}.$$\n\n## Remarks\n\nThe [[preimages|01AHG4G9RT]] $f_i^{-1}(V)$ make each $f_i$ continuous, and any topology that makes each $f_i$ continuous must contain them; this gives the equivalence of the two forms.\nUniversal property: for a topological space $Z$, a map $g \\colon Z \\to X$ is continuous for the initial topology if and only if $f_i \\circ g$ is continuous for every $i$.\n\nExample: for a subset $X$ of a space $Y$ and the inclusion $\\iota \\colon X \\to Y$, the initial topology is the [[subspace topology|0WS9P12FQR]].\nIts open sets are the sets $\\iota^{-1}(V) = X \\cap V$ with $V$ open in $Y$.\nExample: for a product $X = \\prod_i Y_i$ and the projections, the initial topology is the [[product topology|0GBG8B49AA]].\n\nNon-example: the discrete topology on $\\R$ makes the identity map to $\\R$ with its usual topology continuous, but it is not the initial topology for this map.\nIt is strictly finer than needed: the initial topology here is the usual topology, in which $\\set{0}$ is not open.\n\nThe initial topology is also called the weak topology or the induced topology for the family.\n",
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