On Stone's representation theorem

On Stone's representation theorem

Given a compact Hausdorff space X, we consider its regular open algebra, i.e. the Boolean algebra of all regular open subsets of X, RO(X). By Stone’s Representation Theorem, every Boolean algebra A is isomorphic to the clopen algebra of S(A), where S(A) denotes the Stone space of A. This thesis p...

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Bibliographic Details
Main Author: Loo, Dong Lin
Other Authors: Tang Wee Kee
Format: Final Year Project
Language:English
Published: Nanyang Technological University 2022
Subjects:
Science::Mathematics
Online Access:https://hdl.handle.net/10356/156902
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Institution: Nanyang Technological University
Language: English
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Summary:Given a compact Hausdorff space X, we consider its regular open algebra, i.e. the Boolean algebra of all regular open subsets of X, RO(X). By Stone’s Representation Theorem, every Boolean algebra A is isomorphic to the clopen algebra of S(A), where S(A) denotes the Stone space of A. This thesis presents characterisations of the following three items: isomorphism of RO(X), Boolean embedding of RO(X) and quotient spaces of S(RO(X)). Analogues of the last two characterisations in the space of continuous real-valued functions on S(RO(X)) are presented. Some of our results are new and generalize previous known results.

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