I'm working purely on intuition here. The Jordan curve theorem states that a Jordan curve separates the plane into a bounded component and an infinite component. For toy curves, it seems like this bounded component is always open. But in pathological cases, like an Osgood curve which has positive measure, clearly the inside cannot be open since it does not contain an open ball (I think).

Are there examples of Jordan curves with measure $0$ that don't have an open inside? Do Jordan curves with positive measure never have an open inside? More importantly, if the inside is open, does it guarantee that the curve is "non-pathological"?

EDIT: Perhaps my intuition was wrong. According to MO user Timothy Chow in this post, "The Jordan curve theorem was strengthened by Schoenflies to the statement that the two components are homeomorphic to the inside and outside of a circle." By Brouwer's invariance of domain theorem, this implies that the inside component of a Jordan curve is open, if I understand everything correctly.