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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]], and give $X \\times [0, 1]$ the [[product topology|0GBG8B49AA]].\nLet $\\sim$ be the [[equivalence relation|0G8VEZ3WTK]] on $X \\times [0, 1]$ generated by the following relations, for all $x, y \\in X$:\n\n1. $(x, 0) \\sim (y, 0)$;\n2. $(x, 1) \\sim (y, 1)$.\n\nThe *suspension* of $X$ is the quotient\n$$SX = (X \\times [0, 1]) / {\\sim},$$\nwith the [[quotient topology|0NQ1H2X495]].\n\n## Remarks\n\nThe two classes $X \\times \\set{0}$ and $X \\times \\set{1}$ are the two cone points of $SX$.\nA continuous map $f \\colon X \\to Y$ induces a continuous map $Sf \\colon SX \\to SY$ by $[x, t] \\mapsto [f(x), t]$.\n\nExample: let $X = S^0 = \\set{-1, 1}$, a two-point discrete space.\nThen $X \\times [0, 1]$ is two segments, and the relation joins their lower ends and their upper ends.\nThe result is [[homeomorphic|0QT00PB5G2]] to the circle $S^1$.\nMore generally, $S(S^n)$ is homeomorphic to $S^{n+1}$ for every $n \\ge 0$.\n\nNon-example: collapsing only $X \\times \\set{1}$ gives the cone $CX$, not the suspension.\nFor $X = S^1$ the cone is a disc, which has a boundary circle, while $SS^1$ is a sphere.\n\nFor a pointed space $(X, x_0)$, the reduced suspension $\\Sigma X$ also collapses the segment $\\set{x_0} \\times [0, 1]$.\n",
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