My question is about a simple closed curve that is also a space-filling curve. The figure shows 6 iterations of the formation of a Hilbert curve (limit), whose trace is a solid square. I think we may, at each iteration, connect the endpoints of the curve in order to obtain a Jordan curve (simple closed curve), preserving the limit of this sequence of curves (a solid square). So, at the limit, we will have a space-filling, simple closed curve. By the Jordan curve theorem, every simple closed curve "divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points, so that every continuous path connecting a point of one region to a point of the other intersects with that loop somewhere" (Wikipedia).
Question: Does there really exist a space-filling, simple closed curve? What is the interior region of a space-filling, simple closed curve? The empty set?