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  "body": "\n## Definition\n\nLet $X$ and $Y$ be topological spaces and let $I = [0, 1]$.\nThe *join* of $X$ and $Y$ is the quotient space\n$$X * Y = (X \\times Y \\times I) / \\sim,$$\nwhere $X \\times Y \\times I$ has the [[product topology|0GBG8B49AA]], and $X * Y$ carries the [[quotient topology|0NQ1H2X495]].\nThe relation $\\sim$ is the equivalence relation generated by two rules.\nFirst, $(x, y, 0) \\sim (x, y', 0)$ for all $x \\in X$ and $y, y' \\in Y$.\nSecond, $(x, y, 1) \\sim (x', y, 1)$ for all $x, x' \\in X$ and $y \\in Y$.\n\n## Remarks\n\nThe class of $(x, y, t)$ is often written $(1 - t) x + t y$, since it lies on the segment from $x$ to $y$.\nThe images of $t = 0$ and $t = 1$ are copies of $X$ and of $Y$ in $X * Y$.\nExample: if $Y$ is one point, then $X * Y$ is the cone on $X$.\nExample: if $Y = S^0$ is two points, then $X * S^0$ is the unreduced [[suspension|Q7RFNJH76Q]] of $X$.\nExample: $S^0 * S^0$ is homeomorphic to the circle $S^1$, a square with four vertices.\nNon-example: the product $X \\times Y \\times I$ with no identification is not the join; for $X = Y = S^0$ it is four disjoint segments, while $S^0 * S^0$ is a circle.\n",
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