The continuum as a formal space

A constructive definition of the continuum based on formal topology is give n and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include element ary proofs of the Baire and Canter theorems. From a classical standpoint, f ormal reals are seen to be equivalent to the usual reals. Lastly, the relat ion of real numbers as a formal space to other approaches to constructive r eal numbers is determined.