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LatticeValued TFilters and Induced Structures
 Date Issued:
 2019
 Abstract/Description:
 A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a Tconvergence structure which is defined in terms of Tfilters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional.Subcategories of compact and of complete spaces are investigated. It is shown that each Tconvergence space has a compactification with the extension property provided the frame is a Boolean algebra. TCauchy spaces are defined and sufficient conditions for the existence of a completion are given. Tuniform limit spaces are also defined and their completions are given in terms of the TCauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed Tconvergence space, under suitable conditions, it is shown that there exists an order preserving bijection between the set of all strict, regular, Hausdorff compactifications and the set of all totally bounded TCauchy spaces which induce the fixed space.
Title:  LatticeValued TFilters and Induced Structures. 
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Name(s): 
Reid, Frederick, Author Richardson, Gary, Committee Chair Brennan, Joseph, Committee Member Han, Deguang, Committee Member Lang, SheauDong, Committee Member University of Central Florida, Degree Grantor 

Type of Resource:  text  
Date Issued:  2019  
Publisher:  University of Central Florida  
Language(s):  English  
Abstract/Description:  A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a Tconvergence structure which is defined in terms of Tfilters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional.Subcategories of compact and of complete spaces are investigated. It is shown that each Tconvergence space has a compactification with the extension property provided the frame is a Boolean algebra. TCauchy spaces are defined and sufficient conditions for the existence of a completion are given. Tuniform limit spaces are also defined and their completions are given in terms of the TCauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed Tconvergence space, under suitable conditions, it is shown that there exists an order preserving bijection between the set of all strict, regular, Hausdorff compactifications and the set of all totally bounded TCauchy spaces which induce the fixed space.  
Identifier:  CFE0007520 (IID), ucf:52586 (fedora)  
Note(s): 
20190501 Ph.D. Sciences, Mathematics Doctoral This record was generated from author submitted information. 

Subject(s):  Fuzzy Topology  Latticevalued Topology  Fuzzy Sets  LSets  Latticevalued Sets  Fuzzy Filters  Latticevalued Filters  TFilters  Fuzzy Ultrafilters  TUltrafilters  Latticevalued Ultrafilters  Fuzzy Closure  Closure  Regularity  Topological  Pretopological  Compactification  Completion  Selection Maps  Fuzzy Convergence Spaces  TConvergence Spaces  Latticevalued Convergence Spaces  Fuzzy Cauchy Spaces  TCauchy Spaces  Latticevalued Cauchy Spaces  Fuzzy Limit Space  TLimit Spaces  Latticevalued Limit Spaces  Fuzzy Uniform Limit Spaces  TUniform Limit Spaces  Latticevalued Uniform Limit Spaces  Stratified Convergence Spaces  Stratified Filters  Stratified Ultrafilters  Strict Regular Harsdorf Embedding  Category  Strongly Cartesian Closed  Extensional  Topological Constructs  Topological Universe  Initial Structure  Final Structure  Reflective  Bicoreflective  
Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0007520  
Restrictions on Access:  public 20190515  
Host Institution:  UCF 