Applications of the theory of Boolean rings to general topology.

*(English)*Zbl 0017.13502The author applies his theory of the algebra of classes [Trans. Am. Math. Soc. 40,
37–111 (1936; Zbl 0014.34002)] to general topology.

He first establishes a one-one correspondence between “Boolean” rings \(R\) (i. e., rings whose elements are idempotent) and “Boolean” spaces \(H\) (i. e., totally disconnected locally bicompact Hausdorff spaces). The ideals \(J\) of \(R\) correspond to the open sets of \(H\); the quotient-rings \(R/J\) to the complementary closed sets; the principal ideals of \(R\) to the bicompact open sets of \(H\); the prime ideals of \(R\) to the complements of points in \(H\); automorphisms of \(R\) to self-homeomorphisms of \(H\). \(H\) is bicompact if and only if \(R\) has a unit. The “universal” Boolean spaces of given character \(\mathfrak c\) introduced by Tychonoff (Tikhonov), correspond to the “free” Boolean rings generated by \(\mathfrak c\) symbols.

He then discusses “maps” of general spaces \(S\) on Boolean spaces \(H\): points in \(S\) become closed sets in \(H\). He shows that any \(T_0\)-space is characterized topologically by its maps. He uses this fact to discuss the embedding of \(T_0\)-spaces as dense subsets in bicompact spaces (the “problem of extension”), obtaining new results.

He also studies various regularity and normality conditions on \(T_0\)-spaces, among them a new condition of “semi-regularity”, important for his theory of mapping. And finally, he discusses the “function-ring” of all continuous, bounded functions on an arbitrary \(T_0\)-space. He uses this to obtain a solution of the problem of extension.

He first establishes a one-one correspondence between “Boolean” rings \(R\) (i. e., rings whose elements are idempotent) and “Boolean” spaces \(H\) (i. e., totally disconnected locally bicompact Hausdorff spaces). The ideals \(J\) of \(R\) correspond to the open sets of \(H\); the quotient-rings \(R/J\) to the complementary closed sets; the principal ideals of \(R\) to the bicompact open sets of \(H\); the prime ideals of \(R\) to the complements of points in \(H\); automorphisms of \(R\) to self-homeomorphisms of \(H\). \(H\) is bicompact if and only if \(R\) has a unit. The “universal” Boolean spaces of given character \(\mathfrak c\) introduced by Tychonoff (Tikhonov), correspond to the “free” Boolean rings generated by \(\mathfrak c\) symbols.

He then discusses “maps” of general spaces \(S\) on Boolean spaces \(H\): points in \(S\) become closed sets in \(H\). He shows that any \(T_0\)-space is characterized topologically by its maps. He uses this fact to discuss the embedding of \(T_0\)-spaces as dense subsets in bicompact spaces (the “problem of extension”), obtaining new results.

He also studies various regularity and normality conditions on \(T_0\)-spaces, among them a new condition of “semi-regularity”, important for his theory of mapping. And finally, he discusses the “function-ring” of all continuous, bounded functions on an arbitrary \(T_0\)-space. He uses this to obtain a solution of the problem of extension.

Reviewer: Garrett Birkhoff (Cambridge, USA)

##### MSC:

54-XX | General topology |