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 On certain rings of evalued continuous functions
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On certain rings of evalued continuous functions Chew, KimPeu
Abstract
Let C(X,E) denote the set of all continuous functions from a topological space X into a topological space E. R. Engelking and S. Mrowka [2] proved that for any Ecompletely regular space X [Definition 1.1], there exists a unique Ecompactification [formula omitted] [Definitions 2.1 and 3.1] with the property that every function f in C(X,E) has an extension f in [formula omitted]. It is proved that if E is a (*)topological division ring [Definition 55] and X is an Ecompletely regular space, then [formula omitted] is the same as the space of all Ehomomorphisms [Definition 5.3] from C(X,E) into E. Also, we establish that if E is an Htopological ring [Definition 6.1] and X, Y are Ecompact spaces [Definition 2.1], then X and Y are homeomorphic if, and only if, the rings C(X,E) and C(Y,E) are Eisomorphic [Definition 5.3]. Moreover, if t is an Eisomorphism from C(X,E) onto C(Y,E) then [formula omitted] is the unique homeomorphisms from Y onto X with the property that [formula omitted] for all f in C(X,E), where π is the identity mapping on X and t is a certain mapping induced by t. In particular, the development of the theory of C(X,E) gives a unified treatment for the cases when E is the space of all real numbers or the space of all integers. Finally, for a topological ring E, the bounded subring C*(X,E) of C(X,E) is studied. A function f in C(X,E) belongs to C*(X,E) if for any Oneighborhood U in E, there exists a 0neighborhood V in E such that f[X]•V c U and V•f[X] c U. The analogous results for C*(X,E) follow closely the theory of C(X,E); namely, for any E*completely regular space X [Definition 9.5], there exists an E*compactification [formula omitted] of X such that every function f in C (X,E) has an extension f in [formula omitted] when E is the space of all nationals, real numbers, complex numbers, or the real quaternions, [formula omitted] is just the space of all Ehomomorphisms from C*(X,E) into E. This is also valid for a topological ring E which satisfies certain conditions. Also, two E*compact spaces [Definition 10.1] X and Y are homeomorphic if, and only if, the rings C*(X,E) and C*(Y,E) are Eisomorphic, where E is any H*topological ring [Definition 12.8].
Item Metadata
Title 
On certain rings of evalued continuous functions

Creator  
Publisher 
University of British Columbia

Date Issued 
1969

Description 
Let C(X,E) denote the set of all continuous functions
from a topological space X into a topological space E.
R. Engelking and S. Mrowka [2] proved that for any Ecompletely
regular space X [Definition 1.1], there exists a unique Ecompactification
[formula omitted] [Definitions 2.1 and 3.1] with the property
that every function f in C(X,E) has an extension f in [formula omitted].
It is proved that if E is a (*)topological division
ring [Definition 55] and X is an Ecompletely regular space,
then [formula omitted] is the same as the space of all Ehomomorphisms
[Definition 5.3] from C(X,E) into E. Also, we establish that
if E is an Htopological ring [Definition 6.1] and X, Y are
Ecompact spaces [Definition 2.1], then X and Y are homeomorphic
if, and only if, the rings C(X,E) and C(Y,E) are Eisomorphic
[Definition 5.3]. Moreover, if t is an Eisomorphism from
C(X,E) onto C(Y,E) then [formula omitted] is the unique homeomorphisms
from Y onto X with the property that [formula omitted] for all
f in C(X,E), where π is the identity mapping on X and t
is a certain mapping induced by t. In particular, the development
of the theory of C(X,E) gives a unified treatment for the cases
when E is the space of all real numbers or the space of all
integers.
Finally, for a topological ring E, the bounded subring
C*(X,E) of C(X,E) is studied. A function f in C(X,E) belongs
to C*(X,E) if for any Oneighborhood U in E, there exists a 0neighborhood V in E such that f[X]•V c U and V•f[X] c U.
The analogous results for C*(X,E) follow closely the theory of
C(X,E); namely, for any E*completely regular space X
[Definition 9.5], there exists an E*compactification [formula omitted] of
X such that every function f in C (X,E) has an extension f
in [formula omitted] when E is the space of all nationals, real numbers,
complex numbers, or the real quaternions, [formula omitted] is just the space
of all Ehomomorphisms from C*(X,E) into E. This is also valid
for a topological ring E which satisfies certain conditions. Also,
two E*compact spaces [Definition 10.1] X and Y are homeomorphic
if, and only if, the rings C*(X,E) and C*(Y,E) are Eisomorphic, where E is any H*topological ring [Definition 12.8].

Genre  
Type  
Language 
eng

Date Available 
20120307

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 
10.14288/1.0080624

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
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.