Let P be a polynomial functor from Top to Top, by which I mean a functor of the form P(X) = ∐_{i ≥ 0} S_{i} × X^{i} where the S_{i} are finite sets, all but finitely many of which are empty. Corresponding to P there is an ordinary polynomial p(x) = Σ_{i ≥ 0} |S_{i}| x^{i}. I believe that there is a final P-coalgebra X_{P} in Top; the structure map X_{P} -> P(X_{P}) is a homeomorphism; the underlying set of X_{P} is the final P-coalgebra in Set; and a basis for the open sets of X_{P} is given by the preimages of the points of the targets of the maps X_{P} -> P^{(n)}(X_{P}) -> P^{(n)}(•) as n ranges over all nonnegative integers.

Examples are the Cantor space, corresponding to p(x) = 2x, and the space of binary trees, corresponding to p(x) = x^2 + 1.

My question is: if P and Q are two polynomial functors, such that neither p(x) nor q(x) is of the form x + k, and the spaces X_{P} and X_{Q} are homeomorphic, does it follow that the polynomials p(x) - x and q(x) - x have a common root?

More generally, consider all the spaces that can be formed starting with the collection of spaces X_{P} (p(x) not of the form x + k) by taking disjoint unions and products. Can I assign an algebraic number to each of these spaces in a way which is a homeomorphism invariant and commutes with sums and products?