I was thinking about letting be the space consisting of all nonconstant sequences with either 0 or 1 as the elements under the discrete topology. (Note that it is T4). This space definitely has the same cardinality as the continuum, .
Let A be the singleton and B be the singleton . By Urysohn's Lemma, we can find a continuous function with and .
I was hoping that this would establish some kind of bijection between and , but I'm not entirely sure that it is correct. But maybe I gave you some ideas. (And I suppose that it is better than receiving no responses on your post.)