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  "body": "\n## Definition\n\nA *pointed space* is a pair $(X, x_0)$ where $X$ is a [[topological space|0CQHFPGJ45]] and $x_0 \\in X$.\nThe point $x_0$ is the *base point*.\nA *pointed map* $f \\colon (X, x_0) \\to (Y, y_0)$ is a [[continuous map|0VGFHTM75Y]] $f \\colon X \\to Y$ with $f(x_0) = y_0$.\n\n## Remarks\n\nPointed spaces and pointed maps form a category, often written $\\mathbf{Top}_*$.\nA pointed homotopy between pointed maps is a homotopy relative to the base point, and $[X, Y]_*$ denotes the set of pointed homotopy classes.\nExample: $(S^n, e_1)$ with $e_1 = (1, 0, \\ldots, 0)$ is a pointed space.\nNon-example: the empty space has no point, so it carries no structure of a pointed space.\nEvery space $X$ gives a pointed space $X_+ = X \\sqcup \\set{*}$ with a new base point $*$.\n",
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