Hello. Forgive the basic 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.
Are the following conditions equivalent?
1. $\mathrm{T}_pX$ is a linear subspace of $\mathbb R^n$ of dimension $\dim_pX$.
2. We have the following equality, where the limit is taken in a translated neighborhood in $X$ of $p$. $$\lim_{h\to 0} \frac{\pi_{(\mathrm T_pX)^\perp}(h)}{\|…