有關topology的問題

Any simple continuous closed loop on a 2-dimensional plane is a Jordan curve, i.e. a continuous injective function from S^1 to R^2, where S^1 is the unit circle. One can also define S^1 by the closed interval [0,1] by identifying 0 and 1. This means the curve has the same end points at 0 and 1. The Jordan curve theorem states that any Jordan curve bounds an open interior. Precisely this means the compliment of the curve in R^2 consists of two open connected components, and one of them is bounded.

A homeomorphism is a bijective bi-continuous function between two topological spaces. That is, the function has an inverse (i.e. injective and surjective) and they send open sets to open sets between the domain space and target space.