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  "body": "\n## Summary\n\nLet $A$ be a subspace of a space $X$.\nThe pair $(X, A)$ has the homotopy extension property when each [[homotopy|9YA6S6HD9V]] on $A$ extends to all of $X$, once its start on $X$ is given.\nPairs with this property let one move maps on a subspace without losing control of the whole space.\nIt holds for CW pairs and is the defining property of [[cofibrations|W5WS5EY8ZM]].\n\n## Explanations\n\n### L1 human\n\nSuppose you have a map on a big space $X$ and you start to wiggle it on a small part $A$.\nThe homotopy extension property says that you can wiggle the map on the rest of $X$ in a way that matches.\nFor example, a disk with its boundary circle has this property: any deformation of a map on the circle spreads into the disk.\nBadly attached subspaces, such as a limit point of a sequence of isolated points, can fail it.\n",
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