Consider the subspace of of all sequences such that . It is easy to see that is compact, since it is closed in and totally bounded.
On the other hand, for any the convex hull of the points of the form with and is contained in . In other words, we can embed a -simplex in for all , so has infinite topological dimension, and therefore does not embed in .
Consider the countable product of copies of . 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.)
The Noriega Book 
Consider the subspace
of 
of all sequences 
such that ![x_n \in [0, 1/n]](https://s0.wp.com/latex.php?latex=x_n+%5Cin+%5B0%2C+1%2Fn%5D&bg=ffffff&fg=333333&s=0&c=20201002)
. It is easy to see that 
is compact, since it is closed in 
and totally bounded.
On the other hand, for any
the convex hull of the points of the form 
with 
and 
is contained in 
. In other words, we can embed a 
-simplex in 
for all 
, so 
has infinite topological dimension, and therefore does not embed in 
.
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Consider the countable product of copies of![[0,1]](https://s0.wp.com/latex.php?latex=%5B0%2C1%5D&bg=ffffff&fg=333333&s=0&c=20201002)
. 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.)
LikeLiked by 1 person