Consider a (locally closed if needed) locally Euclidean subset $X\subset \mathbb R^n$. Given a point $p\in X$, let $\dim_pX$ denote the dimension of $X$ at $p$ and consider the following condition:
- There are differentiable/smooth germs of paths in $X$ based at $p$ whose tangents at $p$ span a space of dimension $\dim_pX$.
I am hopeful this condition may express that $X\subset\mathbb R^n$ is smooth at $p$ by asserting it has a tangent space with the correct dimension.
If all points of a (locally closed if needed) locally Euclidean subset $X\subset\mathbb R^n$ satisfy the above condition, is $X\subset\mathbb R^n$ an embedded submanifold? Does this characterize submanifolds embedded in Euclidean space?
The question is motivated by thinking of manifolds as "things with tangent spaces".