Question

Suppose that M is compact and that f : M → N is continuous, one-to-one, and onto. Prove that f is a homeomorphism.

Answer 1

To prove that f is a homeomorphism, we need to show that it is a continuous, one-to-one, and onto function.

Step-by-step explanation:

To prove that f is a homeomorphism, we need to show that it is a continuous, one-to-one, and onto function. Since M is compact and f is continuous, we know that the image of M under f is also compact.

Next, we can show that f is one-to-one by assuming that there are two distinct points in M that map to the same point in N. Since f is continuous and one-to-one, this cannot happen.

Finally, we can show that f is onto by assuming that there exists a point in N that does not have a pre-image in M. Again, by continuity, this cannot happen. Therefore, f is a homeomorphism.

Since M is compact and f is continuous, we know that the image of M under f is also compact. Next, we can show that f is one-to-one by assuming that there are two distinct points in M that map to the same point in N.

Finally, we can show that f is onto by assuming that there exists a point in N that does not have a pre-image in M. Therefore, f is a homeomorphism.