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  "body": "\n## Definition\n\nLet $X$ be a topological space and let $A \\subseteq X$ carry the [[subspace topology|0WS9P12FQR]].\nThe subspace $A$ is a *deformation retract* of $X$ if there is a [[homotopy|9YA6S6HD9V]] $H \\colon X \\times [0, 1] \\to X$ such that:\n\n1. $H(x, 0) = x$ for all $x \\in X$;\n2. $H(x, 1) \\in A$ for all $x \\in X$;\n3. $H(a, 1) = a$ for all $a \\in A$.\n\nThe subspace $A$ is a *strong deformation retract* of $X$ if moreover $H(a, t) = a$ for all $a \\in A$ and $t \\in [0, 1]$.\n\n## Remarks\n\nThe map $r = H(\\cdot, 1) \\colon X \\to A$ is a retraction, and the inclusion $i \\colon A \\to X$ is a [[homotopy equivalence|XZG6MTSWCN]] with homotopy inverse $r$.\nSome textbooks, such as Hatcher's, use the name deformation retraction for the strong version.\nExample: $S^{n-1}$ is a strong deformation retract of $\\R^n \\setminus \\set{0}$, via $H(x, t) = (1 - t) x + t x / \\norm{x}$; clause 3 holds since $\\norm{a} = 1$ for $a \\in S^{n-1}$.\nNon-example: a point $p$ of the circle $S^1$ is not a deformation retract of $S^1$, since otherwise $S^1$ would be [[contractible|R5WP8Z69Q8]].\n",
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