Prove that if X is a metrizable topological space and Y is homeomorphic to X, then Y is metrizable
Since X is a metrizable topological space, we have a metric space (X, d).
Let Y be a topological space homeomorphic to X and be a homeomorphism.
Define d' on such that
.
I'll leave it to check d' is indeed a metric.
Since both and are continuous bijection, we see that and are isometries, which implies that an open ball of radius r >0 with respect to a metric d on space X corresponds to an open ball of radius r with respect to a metric d' on space Y, and vice versa. Now, the open balls in Y defined by d' can be given as a basis for a topological space Y. Thus, Y is metrizable.