Constructive irrational space

The Fréchet combination allows the construction of a complete metric on the set of irrational numbers. The constructive study of the resulting spaceM was begun by Errett Bishop. This paper studies the structure ofM in some detail. The constructive approach requires a strong form of the concept of irrational number and particular attention to the distinctions between the various notions of points exterior to a set. The main results are the characterization and construction of all compact and locally compact subspaces ofM.