# When does intersection of measure 0 implies interior-disjointness?

If there are two “nice” shapes in $R^2$, such as circles or polygons, whose intersection has area 0, then they must be interior-disjoint, as their intersection can only contain pieces of their boundary.

My question is: what is a simple term for those “nice” subsets of $R^2$ for which intersection of area 0 implies interior-disjointness?

I am not looking for the most general term – just a simple term which I can use in a paper on another topic.

#### Solutions Collecting From Web of "When does intersection of measure 0 implies interior-disjointness?"

This holds for arbitrary (measurable) subsets $A,B \subset \Bbb{R}^2$, because

$$A^\circ \cap B^\circ \subset A \cap B$$

is an open set (finite intersection of open sets) of measure zero, thus empty, where I denoted the (topological) interior relative to $\Bbb{R}^2$ of a set $M$ by $M^\circ$.