Abstract
We define the continuum up to order isomorphism (and hence homeomorphism) as the final coalgebra of the functor X·ω, ordinal product with ω. This makes an attractive analogy with the definition of the ordinal ω itself as the initial algebra of the functor 1;X, prepend unity, with both definitions made in the category of posets. The variants 1; (X·ω), Xo·ω, and 1;(Xo·ω) yield respectively Cantor space (surplus rationals), Baire space (no rationals), and again the continuum as their final coalgebras.
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