Spooky geometry II

Find a compact metric space that does not embed in \Bbb{R}^n for any n.

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Spooky geometry II

2 thoughts on “Spooky geometry II

  1. Consider the subspace X of \ell^\infty of all sequences (x_n) such that x_n \in [0, 1/n]. It is easy to see that X is compact, since it is closed in \ell^\infty and totally bounded.

    On the other hand, for any k\in \Bbb{N} the convex hull of the points of the form e_j/k with j\in\{1,\dots k\} and 0 is contained in X. In other words, we can embed a k-simplex in X for all k, so X has infinite topological dimension, and therefore does not embed in \Bbb{R}^n.

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  2. Consider the countable product of copies of [0,1]. This is metrizable and compact, but doesn’t embbed in any Euclidean space by the same argument above. (This is homeomorphic to the example given above, in fact.)

    Liked by 1 person

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