On coalgebra of real numbers

We deene the continuum up to order isomorphism (and hence homeomorphism) as the nal coalgebra of the functor X !, ordinal product with !. This makes an attractive analogy with the deenition of the ordinal ! itself as the initial algebra of the functor 1; X, prepend unity, with both deenitions made in the category of posets. The variants 1; (X !), X o !, and 1; (X o !) yield respectively Cantor space (surplus rationals), Baire space (no rationals), and again the continuum as their nal coalgebras.