This is a crosspost of this MSE question.
Given a locally Euclidean (locally homeomorphic to some Euclidean space) topological subspace $X\subset\mathbb R^n$ and $p\in X$, let $\mathrm{T}_pX$ denote the tangent set of $X$ at $p$, namely the set of derivatives of curves in $X$ based at $p$ and differentiable at zero.
Questions.
- Are the following conditions equivalent? What are the implications between them?
- Does the second condition imply $\pi_V$ is a local homeomorphism about $p$?
- Does the first condition naturally furnish a local homeomorphism at $p$ between a neighborhood of $p$ in $\mathrm T_pX$ and a neighborhood of $p$ in $X$?
(For 3, the only natural candidate I see here is orthogonal projection. I don't know how to prove it's locally injective. The conditions of the everywhere differentiable inverse function theorem may be false as exemplified e.g by $f(x)=(x^5\cos(\frac 1x))^{\frac13}$, so that won't work.)
- $\mathrm{T}_pX$ is a linear subspace of $\mathbb R^n$ of dimension $\dim_pX$.
- There's a linear subspace $V\subset\mathbb R^n$ of dimension $\dim_pX$ for which we have the following equality, where the limit is taken in a translated neighborhood in $X$ of $p$. $$\lim_{h\to 0} \frac{\pi_{V^\perp}(h)}{\|h\|}=0$$ (Here $\pi_{V^\perp}$ is the orthogonal projection onto the orthogonal complement of $V\subset\mathbb R^n$.)