What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves?
This set contains simply-connected polygons and circles but also polygons with holes. It doesn't include polygons with a singular point added or missing.
MOTIVATION: I am looking for a set, as large as possible, on which the symmetric difference defines a metric, i.e., the symmetric difference of two different shapes always has positive area. This holds (I think) for the set of closed simply-connected 2-dimensional shapes, but I am trying to expand it further, to shapes that may have holes. The set of all closed connected shapes is too inclusive as it includes, for example, both the unit disc $D$ and $D\setminus\{(0,0)\}$, and the difference between them has area 0.
The term would probably be 2-dim sub-manifold, but, given your motivation, I don't see why you need to limit yourself to those. The Lebesgue measure of the symmetric difference of measurable sets defines a metric (where sets which differ by a set of measure $0$ are considers identical).
You can, of course, limit your metric space to "nicer" sets (e.g., open sets) but your metric space will not be complete: Consider a Sierpinski carpet $S$ (compact, empty interior) of non-zero measure (e.g. the Wallis Sieve). One can easily construct a sequence of nested open sets $A_n\subset A_m$ for $n>m$ so that $\cap A_n=S$, so $\lim_n \lambda(A_n\Delta S)=0$ but there is no open set $A$ such that $\lambda(A\Delta S)=0$.