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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].\nA function $f \\colon X \\to Y$ is a topological embedding when $f$ is [[injective|0PVG1CRAJN]] and the map $X \\to f(X)$, $x \\mapsto f(x)$, is a [[homeomorphism|05GZ11PWRY]].\nHere the [[image|0Q3F20XB0A]] $f(X)$ has the [[subspace topology|0D5YHQGDJY]] of $Y$.\n\n## Remarks\n\nEach embedding is continuous, because the [[inclusion of a subspace is continuous|065J822CKN]].\nThe inclusion of a subspace is an embedding. The map $\\mathbb{R} \\to \\mathbb{R}^2$, $x \\mapsto (x, 0)$, is an embedding.\nThe map $t \\mapsto (\\cos t, \\sin t)$ from $[0, 2\\pi)$ to $\\mathbb{R}^2$ is continuous and injective, but not an embedding.\nA continuous injective map from a [[compact space|0QPHW703PV]] to a [[Hausdorff space|0MXB4CKHFW]] is an embedding.\n\n## Explanations\n\n### L1 human\n\nAn embedding puts a copy of $X$ inside $Y$ without changing its shape.\nAn injective continuous map can still bring the ends of the space together in the limit. Then it is not an embedding.\n",
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