# Spooky geometry II

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

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 $\Bbb{R}^n$.
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.)