Wikipedia offers the following definition for an (embedded) submanifold:
An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a topological embedding.
I've been wondering if one could not equivalently define a submanifold like this:
$(\ast)$ Let $M$ be a smooth $m$-manifold and let $N$ be a subset. Then $N$ is an $n$-submanifold if and only if it is itself a $n$-manifold when endowed with the subspace topology.
Is this definition equivalent to the usual definition with embedding?
Edit!
It appears that "my" definition is in fact common, too.
I seem to have found it in this book on page 5:
1.5. Definition. A subset $M \subset \mathbb R^n$ is called differentiable submanifold of $\mathbb R^n$ of dimension $m \le n$ if to each $x\in M$ there corresponds an invertible germ $\widetilde{\phi}:(\mathbb R^n,x) \to (\mathbb R^n,0)$ such that $\widetilde{\phi} (M,x) = (\mathbb R^m,x)\subset (\mathbb R^n,x)$ ($\mathbb R^m$ linearly embedded in $\mathbb R^n$ for $m \le n$)
This is exactly the same as $(\ast)$ if in $(\ast)$ we let $M = \mathbb R^n$, right?