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  "body": "\n## Statement\n\nLet $n \\ge 1$, let $U$ be an [[open subset|02BH3MCBQB]] of [[$\\R^n$|0XPR1E85W0]], and let $f \\colon U \\to \\R^n$ be a [[continuous|0VGFHTM75Y]] [[injective|09NWETVTNY]] map.\nThen $f(U)$ is open in $\\R^n$, and $f$ is a [[homeomorphism|0QT00PB5G2]] from $U$ onto $f(U)$.\n\n## Remarks\n\nFor $n = 1$ the claim follows from the intermediate value theorem, since $f$ is then strictly monotone on each interval in $U$; so let $n \\ge 2$.\nIt suffices to show that, for every closed ball $B \\subseteq U$, the image $f(B \\setminus \\partial B)$ of its interior is open.\nThe image $f(\\partial B)$ is homeomorphic to the sphere $S^{n-1}$, so by the [[Jordan–Brouwer separation theorem|2Q008XWQDE]] its complement has two components, a bounded one and an unbounded one.\nA homology argument shows that $f(B \\setminus \\partial B)$ is connected and disjoint from $f(\\partial B)$ and that $\\R^n \\setminus f(B)$ is connected, which forces $f(B \\setminus \\partial B)$ to be the bounded component, an open set.\nThe second claim follows from the first, since $f$ then maps open subsets of $U$ to open sets.\n\nExample: it implies invariance of dimension, that is, $\\R^m$ and $\\R^n$ are not homeomorphic if $m \u003c n$.\nIndeed, a homeomorphism $\\R^n \\to \\R^m \\subset \\R^n$ would have an open image inside the hyperplane $\\R^m \\times \\set{0}$.\nBut this hyperplane has empty interior in $\\R^n$.\n\nBoth hypotheses are needed.\nWithout injectivity, $f(x) = x^2$ on $\\R$ has image $[0, \\infty)$, which is not open.\nThe target must have the same dimension: the map $x \\mapsto (x, 0)$ from $\\R$ to $\\R^2$ is continuous and injective, but its image is not open in $\\R^2$.\n\n## Explanations\n\n### L1 human\n\nA continuous one-to-one map from an open piece of $n$-dimensional space into $n$-dimensional space cannot squash anything flat or tear it open.\nIts image is again an open piece, and the map can be undone continuously.\n\n### L2 human\n\nThe theorem shows that the dimension of a topological manifold is well defined.\nIt also shows that a continuous bijection between manifolds of the same dimension without boundary is a homeomorphism.\nIt is false in infinite dimensions: on a Hilbert space, the shift operator is a continuous injective linear map whose image is not open.\nIt is usually proved with homology, through the Jordan–Brouwer separation theorem, or with the Brouwer fixed point theorem.\n",
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