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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]], let $A$ be a [[closed|0ZCK8DQYN1]] subset of $Y$, and let $f \\colon A \\to X$ be a [[continuous map|0VGFHTM75Y]].\nLet $\\sim$ be the smallest [[equivalence relation|0G8VEZ3WTK]] on the [[disjoint union|ZXJKSB30WF]] $X \\sqcup Y$ such that $a \\sim f(a)$ for every $a \\in A$.\nThe *adjunction space* is the quotient set\n$$X \\cup_f Y = (X \\sqcup Y) / {\\sim},$$\nwith the [[quotient topology|0NQ1H2X495]].\nThe map $f$ is called the attaching map.\n\n## Remarks\n\nThe composite $X \\to X \\sqcup Y \\to X \\cup_f Y$ is a closed [[topological embedding|00ZG9Y9TR4]], and $Y \\setminus A$ maps homeomorphically onto an open subset.\nMany authors only ask that $A$ be a subspace; the closedness of $A$ is needed for these two facts.\n\nExample: let $Y = D^2$ be the closed unit disc, $A = S^1$ its boundary circle, and $X = S^1$ with $f(z) = z^2$.\nThen $f$ is continuous and $A$ is closed in $Y$, so $X \\cup_f D^2$ is an adjunction space.\nIt is homeomorphic to the real projective plane.\n\nExample: if $X$ is a one-point space, then $X \\cup_f Y$ is the quotient $Y / A$ that collapses $A$ to a point.\n\nNon-example: let $Y = [0, 1]$, let $A = (0, 1]$, let $X$ be a one-point space, and let $f$ be the constant map.\nThe clause that $A$ is closed fails, and the quotient is a two-point space whose open sets are $\\varnothing$, $\\set{[A]}$ and the whole space.\nThere the image of $X$ is the open point $[A]$, which is not closed.\n",
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