Interpreting definitions of a manifold and a sub-manifold

I am given definitions for a sub-manifold and a manifold.

A set $S\subset\Bbb{R^n}$ is an $m$-dimensional submanifold in $\Bbb{R^n}$, if there exists an area $U\subset\Bbb{R^m}$ and a homeomorphism $g:U\to S$

A set $S\subset\Bbb{R^n}$ is an $m$-dimensional manifold in $\Bbb{R^n}$, if for every $x\in S$ there is a relatively open set $S_x$ in $S$ such that $x\in S_x$ and $S_x$ is a $m$-dimensional submanifold.

The second definition is very confusing. However, I think the second definition means, that if we can show some set $S$ to be a submanifold, then $S$ must also be a manifold in its own right. Is this correct?

For example let $f:(a,b) \to \Bbb{R}$ be a continuous function. It can be shown ( rather easily ) that by the first definition the graph surface of $f$ is a sub-manifold. Then by the second definition I can directly claim that the graph surface is also a manifold, right?

Solution 1:

Using your definition, it is trivial to see that every $m$-submanifold in $\mathbb R^{n}$ is also a manifold in $\mathbb R^n$.

But there are manifolds in $\mathbb R^n$ that is not a submanifold. For example, the sphere $\mathbb S^m = \{ (x_1, \cdots, x_{m+1}) \in \mathbb R^{m+1}: x_1^2 + \cdots x_{m+1}^2 = 1\}$ is a $m$-manifold in $\mathbb R^{m+1}$, but not a submanifold.

As a remark, your definition of submanifolds/manifolds are quite uncommon.