For sure every curve that is homeomorphic to a circle is a simple closed curve, but is every simple closed curve homeomorphic to a circle?
Is there a proof for that, or is there some topological invariant that is not not shared with simple closed curve and a curve that is homeomorphic to a circle?
"Simple closed curve" = "non-self-intersecting continuous closed curve in plane."
"non-self-intersecting continuous closed curve in plane" is the same as "image of a continuous injective function from the circle to the plane".
Since the circle is compact, such a function has a continuous inverse and so is a homeomorphism between the circle and the curve.