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 Admissible subrings of realvalued continuous functions
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Admissible subrings of realvalued continuous functions Choo, EngUng
Abstract
The object of this thesis is to study the relations between the algebraic properties of admissible subrings [Definition 0.4] of the ring C(X) of all realvalued continuous functions on X and the topological properties of the space X. Given an admissible subring G of C(X), there exists a unique G*compactification [Definition 1.7 and 0.17] Y of X, with the properties that X is G*embedded [Definition 0.16] in Y and G*(Y) [Definition 0.12] is admissible. If the cardinality Y  X of Y  X is finite or dG [Definition 2.1] is finite, then Y — X = dG. From Y, we can get the unique Grealcompactification [Definition 1.8] Z of X where X is Gembedded in Z, G(Z) is admissible and Z is G(Z)realcompact [Definition 1.1]. It is proved that a Grealcompact space X and an Hrealcompact space Y are homeomorphic if G and H are isomorphic admissible subrings. Let D(X) be the subring of all closed bounded functions in C(X). Then X is countably compact iff D(X) is admissible and closed under uniform convergence. For any admissible subring G of C(X), if dG is finite, then dG ≤ dD(X). Let B(X) be the subring of all bounded functions in C(X) which niap zero sets to closed sets in R. Then X is pseudocompact iff B(X) is admissible and closed under uniform convergence.
Item Metadata
Title 
Admissible subrings of realvalued continuous functions

Creator  
Publisher 
University of British Columbia

Date Issued 
1971

Description 
The object of this thesis is to study the relations between the algebraic properties of admissible subrings [Definition 0.4] of the ring C(X) of all realvalued continuous functions on X and the topological properties of the space X.
Given an admissible subring G of C(X), there exists a unique G*compactification [Definition 1.7 and 0.17] Y of X, with the properties that X is G*embedded [Definition 0.16] in Y and G*(Y) [Definition 0.12] is admissible. If the cardinality Y  X of Y  X is finite or dG [Definition 2.1] is finite, then Y — X = dG. From Y, we can get the unique Grealcompactification [Definition 1.8] Z of X where X is Gembedded in Z, G(Z) is admissible and Z is G(Z)realcompact [Definition 1.1]. It is proved that a Grealcompact space X and an Hrealcompact space Y are homeomorphic if G and H are isomorphic admissible subrings.
Let D(X) be the subring of all closed bounded functions in C(X). Then X is countably compact iff D(X) is admissible and closed under uniform convergence. For any admissible subring G of C(X), if dG is finite, then dG ≤ dD(X). Let B(X) be the subring of all bounded functions in C(X) which niap zero sets to closed sets in R. Then X is pseudocompact iff B(X) is admissible and closed under uniform convergence.

Genre  
Type  
Language 
eng

Date Available 
20110330

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080480

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

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Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.