Mathematical Sciences: RUI: Sequences and Compactness       Conditions
{'Code': '03040200', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
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The object of this project is to solve a number of related                 problems concerning convergence of sequences, which have arisen                 during the past twenty years.  The methods to be employed                       include techniques from topology, set theory and mathematical                   logic.  The main problem asks:  Is every product of                             sequentially compact spaces countably compact?  It is one of                    the few remaining, long-standing, unsolved problems concerning                  ultrafilters on the natural numbers.  This simple-sounding                      problem remains unsolved after more than 20 years of work.  A                   related problem, just as old, asks:  Is the product of two                      countably compact topological groups always countably compact?                  Other problems concerning sequences will also be considered.                    For example, does there exist a first countable, countably                      compact separable, non-normal Hausdorff space?                                       The main problem and others to be considered have                          formulations in both set theory and topology, and solutions to                  these problems would advance areas of mathematics where                         convergence of sequences is relevant.  While it is unlikely                     that applied mathematics will benefit immediately,                              clarification of basic issues is always welcome, and a clearer                  view of the real world and its various mathematical models may                  result.
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{'Name': 'University of North Carolina Greensboro', 'CityName': 'GREENSBORO', 'ZipCode': '274125068', 'PhoneNumber': '3363345878', 'StreetAddress': '1000 SPRING GARDEN ST', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'North Carolina', 'StateCode': 'NC', 'CONGRESSDISTRICT': '06', 'CONGRESS_DISTRICT_ORG': 'NC06', 'ORG_UEI_NUM': 'C13DF16LC3H4', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF NORTH CAROLINA AT GREENSBORO', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '1267', 'Text': 'TOPOLOGY'}
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