TITLE: $S^2$ is not a countable union of embedded circles $S^1$
QUESTION [5 upvotes]: I have two questions in point-set topology.

How can I show that $S^2$ is not a countable union of embedded circles $S^1$?
How can I show that if $m>n$ then every nonempty open set $U$ in $\Bbb R^m$ is cannot be contained in a union of finitely many hyperplanes in $\Bbb R^m$ of dimension at most $n$?

I think I have to use some topological invariants, maybe homology, for example. Any hints? Thanks in advance.

REPLY [2 votes]: To add to Henno's nice answer let me point out that you may use homology (the invariance of domain theorem) if for some reason you prefer that to some elementary dimension theory to see that embedded circles are nowhere dense. You can also use measure theory for problem #2 but not for problem #1, as there are Jordan curves with nonzero measure.