## Applications of the theory of Boolean rings to general topology.(English)Zbl 0017.13502

The 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.

### MSC:

 54-XX General topology

### Keywords:

regularity; normality; $$T_0$$-spaces

Zbl 0014.34002
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