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  "body": "\n## Definition\n\nLet $X$ and $Y$ be topological spaces, let $I = [0, 1]$, and let $f \\colon X \\to Y$ be a [[continuous map|0VGFHTM75Y]].\nThe *mapping cylinder* of $f$ is the space\n$$M_f = \\bigl( (X \\times I) \\sqcup Y \\bigr) / \\sim,$$\nwhere $X \\times I$ has the [[product topology|0GBG8B49AA]] and $\\sqcup$ is the [[disjoint union|ZXJKSB30WF]].\nThe relation $\\sim$ is the smallest equivalence relation with $(x, 1) \\sim f(x)$ for all $x \\in X$.\nThe space $M_f$ carries the [[quotient topology|0NQ1H2X495]].\n\n## Remarks\n\nThe map $j \\colon X \\to M_f$, $x \\mapsto [(x, 0)]$, is an embedding, and $Y$ embeds in $M_f$ as well.\nThe map $r \\colon M_f \\to Y$ with $r[(x, t)] = f(x)$ and $r(y) = y$ is a homotopy equivalence, $Y$ is a strong [[deformation retract|CEK4SC20WK]] of $M_f$, and $f = r \\circ j$.\nExample: for $Y$ a point, $M_f$ is the cone $CX = (X \\times I) / (X \\times \\set{1})$.\nExample: for $f = \\id_X$, $M_f$ is homeomorphic to $X \\times I$.\nNon-example: the space $(X \\times I) \\sqcup Y$ without the identification is not $M_f$; for $X$ and $Y$ nonempty it is disconnected even when $X$ and $Y$ are connected, while $M_f$ is then connected.\n",
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