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    Every $m$-manifold imbeds as a closed subspace of $\RR^{2m+1}$

    E-U9VN1

    Area
    Topology
    Topics
    Dimension, Manifolds
    Status
    draft

    Corollary. Every \(m\)-manifold can be imbedded in \(\mathbb{R}^{2m+1}\) as a closed subspace.

    Solution

    An \(m\)-manifold \(M\) is locally compact Hausdorff and second-countable, and every compact subspace of \(M\) has dimension at most \(m\), as shown in the proof of E-PROOB. By E-26ELV, a locally compact Hausdorff space with a countable basis, all of whose compact subspaces have dimension at most \(m\), embeds as a closed subspace of \(\mathbb R^{2m+1}\). Therefore there is an embedding \[ M\hookrightarrow\mathbb R^{2m+1} \] whose image is closed.

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