Boolean Space

• Main Author: Sun, Tzeng-hsiang
• Format: Others
• Published: DigitalCommons@USU 1965
• Subjects: boolean rings; boolean space; lattice; cantor space; Applied Mathematics; Mathematics; Physical Sciences and Mathematics
• Online Access: https://digitalcommons.usu.edu/etd/6670; https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=7860&context=etd

M. H. A. Stone showed in 1937 and subsequently that many interesting and important results of general topology involve latices and Boolean rings. This type of result forms the substance of this thesis. Theorem 4, page 11, states that for any r ≠ 0 in a Boolean ring, there exists a homomorphism h into I2 , (the field of integers modulo 2), such that h(r) = 1. Theorem 3, page 6, states that any subring of a characteristic ring of a Boolean space X is the whole ring if it has the two points property (that is, given x, y in X and a, b in I2, there exists a g such that g(x) = a and g(y) = b). From these two theorems follows the Stone Representation theorem which states that any Boolean ring is isomorphic to the characteristic ring of its Stone space. Theorem 1, page 11, is independent of other theorems. It states that any compact Hausdorff space is the continuous image of some closed subset in a Cantor space. Theorem 5, page 23, states that a topological space can be embedded in a Cantor space as a subspace if and only if it is Boolean. This theorem uses the Dual Representation theorem as its sufficient part. It states that any Boolean space is homomorphic to the Stone space of its characteristic ring.