Definition.Let X and Y be a topological space; let be a bijection.

If both the function h and the inverse function are continuous, then h is called a homeomorphism.

Lemma 1.Let be given by the equation

.

Then g is continuous if and only if the functions

and

are continuous.

( )

Suppose f is a continuous function.

Since f is a well-defined function, it is clear that defined by is a bijection. Let be an open set of relative to the product topology containing an arbitrary point . It follows that

.

Since both an identity map and f are continuous functions, and are open, so is their intersection. It follows that is open. Thus, is continuous.

Now, we need to show that defined by is a continuous function. Let U be an open set containing an arbitrary point of . Then, , where is an intersection of a neighborhood of with . Specifically,

,

where M is a set and V is an open set in Y containing M.

Since has the topology relative to the product topology , is open in . Thus, is continuous.

By the defintion given the above, is a homeomorphism.

( )

Suppose is a homeomorphism.

Since an identity map is continuous and given by is continuous by hypothesis, is continuous by lemma 1.

Note: This is my attemp to this problem. Any feedback or correction is welcomed.