I am trying to work out the details of following example from page 101 of Tu's An Introduction to Manifolds:
Example 9.3. Let $\Gamma$ be the graph of the function $f(x) =
\sin(1/x)$ on the interval $]0, 1[$, and let $S$ be the union of
$\Gamma$ and the open interval $I=\{(0,y)\in \mathbb{R^2} |−1<y<1\}$.The subset $S$ of $\mathbb{R^2}$ is not a regular submanifold for the
following reason: if $p$ is in the interval $I$, then there is no
adapted chart containing $p$, since any sufficiently small
neighborhood $U$ of $p$ in $\mathbb{R^2}$ intersects $S$ in infinitely
many components.
Using Tu's definitions, I need to show that given a neighborhood $U$ of $p$, there exists no homeomorphism $\phi$ from $U$ onto an open set of $V \subset\mathbb{R^2}$ with the property that $U \cap S$ is the pre-image with respect to $\phi$ of the $x$ or $y$ axes intersected with $V$.
I am not sure where the fact that $U \cap S$ has infinitely many components comes into play. I tried to derive a contradiction using this fact; but even if restrict my attention to connected neighborhoods $U$ of $p$, the intersection of the connected set $\phi(U)$ with the $x$ or $y$ axes might have infinitely many components (I think), so there's no contradiction with the fact that homeomorphism preserves components.
I would appreciate any help!
Best Answer
Let $X$ denote the $x$-axis. Assume that $U\ni p$ is open and $\phi(U)$ is an open subset of $\mathbb R^2$ with $\phi:U\to \phi(U)$ a homeomorphism. If $\phi(U)$ intersects $X$ in several components, let $C$ denote the connected component containing $\phi(p)$. Since $\phi(U)$ is open, $\phi(U)\cap X$ is open in $X$. But $X\approx\mathbb R$ is locally connected, therefore components of open subsets are open. So $C$ is open and $X-C$ is closed in $X$. Since $X$ is closed in $\mathbb R^2$, $X-C$ is closed in $\mathbb R^2$ and $\phi(U)-(X-C)$ is open.
Applying $\phi$ we get a homeomorphism $\phi^{-1}(\phi(U)-(X-C))\approx \phi(U)-(X-C)$, the later of which intersects the $x$-axis in just a single component, which is a contradiction.