By now, even a skeptical reader should be thoroughly sold on the utility of embeddings into cubes. But the sales pitch is far from over! We next pursue the consequences of a result stated earlier: If X is completely regular, then Cb(X) completely determines the topology on X. In brief, to know Cb(X) is to know X. Just how far can this idea be pushed? If Cb(X) and Cb(Y) are isomorphic (as Banach spaces, as lattices, or as rings), must X and Y be homeomorphic? Which topological properties of X can be attributed to structural properties of Cb(X) (and conversely)?

These questions were the starting place for Marshall Stone's 1937 landmark paper, “Applications of the Theory of Boolean Rings to General Topology” [140]. It's in this paper that Stone gave his account of the Stone– Weierstrass theorem, the Banach–Stone theorem, and the Stone– Čech compactification. (These few are actually tough to find among the dozens of results in this mammoth 106-page work.) A signal passage from his introduction may be paraphrased as follows: “We obtain a reasonably complete algebraic insight into the structure of Cb(X) and its correlation with the structure of the underlying topological space.” Stone's work proved to be a gold mine – the digging continued for years! – and its influence on algebra, analysis, and topology alike can be seen in virtually every modern textbook.

Independently, but later that same year (1937), Eduard Čech [24] gave another proof of the existence of the compactification but, strangely, credits a 1929 paper of Tychonoff for the result (see Shields [136] for more on this story).