2
$\begingroup$

I always read that an $n$-manifold is a topological space that locally looks like $\mathbb R^n$. But in the definition we require that each point has a neighborhood that is homeomorphic to "an open subset of $\mathbb R^n$" and not necessarily all of $\mathbb R^n$. Knowing that an open subset of $\mathbb R^n$ need not be homeomorphic to $\mathbb R^n$ this seems to be confusing!!

  • 0
    @Pete: Will do.2012-03-18

1 Answers 1

2

Suppose that a point $x\in M$ has a nbhd $U$ homeomorphic to an open subset $V$ of $\Bbb R^n$. Let $h:U\to V$ be a homeomorphism. Since $h(x)\in V$, and $V$ is open in $\Bbb R^n$, there is an $r>0$ such that the open ball $B(h(x),r)\subseteq V$. Note that $B(h(x),r)$ is homeomorphic to $\Bbb R^n$. Now let $B=h^{-1}[B(h(x),r)]$; then $B$ is homeomorphic to $B(h(x),r)$ and hence to $\Bbb R^n$, and clearly $x\in B\subseteq U$. Thus, $x$ has a nbhd homeomorphic to $\Bbb R^n$.

Thus, requiring that a point of $M$ have a nbhd homeomorphic to an open subset of $\Bbb R^n$ is in fact equivalent to requiring that the point have a nbhd homeomorphic to $\Bbb R^n$ itself. In particular, the two definitions of manifold are equivalent.