It is widely known that $C^1$ manifolds are topological spaces locally homeomorphic to Euclidean spaces and possessing $C^1$ chart-converters. They have a tangent space at every point, regarding as the equivalent classes of $C^1$ paths starting from that point under tangency, that approximates smooth "neighbourhood behaviour" by linear ones. It is also well-known that a $C^1$ manifold admits a unique smoothing process.

However, the notion of "tangency" is intuitively not a concept measuring "smoothness". It just says that two curves do not intersect "transversely", which can be precisely described by how fast the distance between a pair of points on them decreases as the points follow curves approaching an intersection. So I came up with the following definition of a **quasi-smooth manifold**, and wonder: **Does this definition include only smooth manifolds, or does it also include exceptional structures?**

Let $(X,d)$ be a second-countable, locally compact and locally path-connected metric space. Suppose that $(X,d)$ has **enough rectifiable paths** (as explained at the end). $\forall p \in X$, let $\Gamma_pX$ denote the space of all rectifiable paths $\gamma$ starting at $p=\gamma(0)$ and parametrized with *constant speed* (the constant path is denoted as $c_p$). Define $T_pX$, the **formal tangent space** at $p$ , to be the space of equivalent classes in $\Gamma_pX$, where
$$\gamma_1 \sim \gamma_2 \iff d(\gamma_1(t),\gamma_2(t)) \sim o(t) \text{ as } t \rightarrow 0^+$$
endowed with a metric
$$\rho ([\gamma_1],[\gamma_2])=\overline{\lim}_{t \rightarrow 0^+} \dfrac{d(\gamma_1(t),\gamma_2(t))}{t}$$
This metric is well-defined since $$d(\gamma_1(t),\gamma_2(t))\, \le\, d(\gamma_1(t),p)+d(\gamma_2(t),p)\, \le\, \mathrm{len}(\gamma_1)|^t_0+\mathrm{len}(\gamma_2)|^t_0\, \le\, Ct$$ for some non-negative constant $C$ by definition. It can be shown that $T_pX$ admits a non-negative homogeneous structure: for $s \ge 0$, let $s[\gamma(t)]:=[\gamma(s \cdot t)] \subseteq T_pX$, then we have equalities similar to the ones in normed vector spaces like $s[c_p]=[c_p]$ and $\rho(s[\gamma_1],s[\gamma_2])=s \rho ([\gamma_1],[\gamma_2])$.

We can also give a global topology $\tau$ on $\bigsqcup_{p \in X}T_pX$. For any open set $U \subseteq X$ and continuous function $f:U \rightarrow \Bbb{R}^+$, let
$$V(U,f)=\{ (p,[\gamma]) 
\ :\ p \in U,\, [\gamma] \in T_pX,\, \rho([\gamma],[c_p])<f(p) \}$$ The topology $\tau$ is generated by such $V$. We call this space the **formal tangent bundle** of $(X,d)$ and denote it as $TX$.

While $T_pX$ has a nice structure, it is too large and does not coincide with the usual tangent space on smooth manifolds. So we define $T^\delta_pX$, the **regular tangent space(s)** at $p$, to be a maximal non-negative homogeneous subspace of $T_pX$ satisfying $$\forall [\gamma_1],[\gamma_2] \in T^\delta_pX,\ d(\gamma_1(t),\gamma_2(t)) \sim Ct+o(t) \text{ as }t \rightarrow 0^+$$ for some constant $C$. This space $T^\delta_pX$ may not be unique and may not be rich in elements (including possibilities when $T^\delta_pX=[c_p]$) but on smooth manifolds it corresponds to the usual tangent space.

Finally, $(X,d)$ is said to be a **quasi-smooth manifold** if there exists a choice of pointwise homeomorphic $T^\delta_pX$ for every $p \in X$ such that $T^\delta_pX$ is also homeomorphic to an open neighbourhood of $p$.

We can also consider a stronger requirement: If $\forall p,q \in X$ (not necessarily different), there exists a rectifiable path $\gamma : (a-\varepsilon,b+\varepsilon) \rightarrow X$ such that $\gamma(a)=p,\gamma(b)=q$ and for any $r=\gamma(t_0) \; (t_0 \in (a,b)), [\gamma(t_0 \pm t)] \in T^\delta_rX$, then we can conclude that $T^\delta_pX$ varies "quasi-smoothly" and we will say $(X,d)$ admits a **regular tangent bundle** (seen as a subspace of $TX$). Would every quasi-smooth manifold $X$ with a regular tangent bundle actually be a smooth manifold?

A continuous map $f:(X,d_X) \rightarrow (Y,d_Y)$ is said to be **quasi-smooth** if $\forall \gamma_1,\gamma_2 \in \Gamma_{p}X \; (p \in X)$, for some constants $C_1$, $C_2$,
$$d_X(\gamma_1(t),\gamma_2(t)) \sim C_1t+o(t) \text{ as } t \rightarrow 0^+$$
$$\implies d_Y(f \circ \gamma_1(t),f \circ \gamma_2(t)) \sim C_2t+o(t) \text{ as } t \rightarrow 0^+$$
It can be shown that quasi-smooth maps have "derivatives" that send regular tangent spaces to regular tangent spaces and thus are homomorphisms between quasi-smooth manifolds, which gives a potential categorical structure on them.

*:$(X,d)$ is said to **have enough rectifiable paths** if $\forall p,q \in X$ and for every countable nowhere-dense set $S \subseteq X$ such that $p,q$ belong to the same path-connected component of $X \backslash S$, there exists a rectifiable path $\gamma \subseteq X \backslash S$ connecting $p,q$. In particular, this property is enough to ensure that given any point and direction at the point, there will be rectifiable paths from $p$ to $q$ avoiding that direction.