I'm specifically looking at this in the context of a continuous closed curve $u:\mathbb{S}^1\rightarrow\mathbb{R}^2$ and a sequence of continuous closed curves $\{u_n\}$ converging in some sense to $u$. Hausdorff convergence seems to be the more intuitive notion for me, defined for two curves $u$ and $v$ by \begin{equation} d_H(u,v)=\inf\{\epsilon>0:u\subseteq\mathcal{N}_\epsilon(v),v\subseteq\mathcal{N}_\epsilon(u)\} \end{equation} where $\mathcal{N}_\epsilon$ is an open $\epsilon$-neighbourhood.
In this way, $u$ and $v$ are close together if every point in $u$ is close to some point in $v$ and vice versa. There is no reference to parametrisation here. But what is uniform convergence of $u_n$ to $u$ in this case? Does this require a parametrisation of each $u_n$ so that we can say \begin{equation} \sup_{x\in\mathbb{S}^1}||u(x)-u_n(x)||\rightarrow0 \end{equation} as $n\rightarrow\infty$? Or is there some way of describing uniform convergence without reference to a parametrisation, in a similar vein to Hausdorff convergence? Does one imply the other? Thanks
Uniform convergence implies Hausdorff (this is immediate) but not the other way around (even for Jordan curves). For example, let $C$ denote the unit circle. For each $n$ take a Jordan curve $C_n$ within Hausdorff distance $\le 1/n$ from $C$ such that $C_n$ winds along $C$ in the positive direction $n$ times and then backtracks and winds in the negative direction $n$ times before closing.