{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/74672"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/74672","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Properties of cocontinuous functions and cocompact spaces","abstract":"In this paper we study the concept of cotopology in the areas of cocon·tinuous functions and cocompact spaces. Initially we investigate and provide needed results concerning closed bases for a topological space. We then study cocontinuous functions by relating them to various other weaker forms of continuous functions, namely c-continuous, almost continuous and weakly continuous. We show that if (Y,U) is locally compact T₂, then f:(X,T)-->(Y,U) is cocontinuous if and only if f⁻¹(0) ε T for every 0 ε U such that (Y - 0) is compact. We note that every almost continuous function is cocontinuous, and we provide conditions under which a weakly continuous function is cocontinuous. We also show that a cocontinuous function from a saturated space to a regular space is continuous. In the area of cocompact spaces we first provide a partial answer to a question of J. M. Aarts as to when the union of cocompact subsets of a space is cocompact. We show that the union of a closed cocompact subset and a closed compact subset is cocompact. We then introduce the properties, locally cocompact and somewhere cocompact, and relate them to property L which was introduced by R. McCoy. We show that every somewhere cocompact regular space has property L, and that every locally cocompact regular space has property L locally. We provide examples to show that neither cocompact nor locally cocompact is equivalent to property L.","abstract_html":"In this paper we study the concept of cotopology in the areas of cocon·tinuous functions and cocompact spaces. Initially we investigate and provide needed results concerning closed bases for a topological space. We then study cocontinuous functions by relating them to various other weaker forms of continuous functions, namely c-continuous, almost continuous and weakly continuous. We show that if (Y,U) is locally compact T₂, then f:(X,T)-->(Y,U) is cocontinuous if and only if f⁻¹(0) ε T for every 0 ε U such that (Y - 0) is compact. We note that every almost continuous function is cocontinuous, and we provide conditions under which a weakly continuous function is cocontinuous. We also show that a cocontinuous function from a saturated space to a regular space is continuous. In the area of cocompact spaces we first provide a partial answer to a question of J. M. Aarts as to when the union of cocompact subsets of a space is cocompact. We show that the union of a closed cocompact subset and a closed compact subset is cocompact. We then introduce the properties, locally cocompact and somewhere cocompact, and relate them to property L which was introduced by R. McCoy. We show that every somewhere cocompact regular space has property L, and that every locally cocompact regular space has property L locally. We provide examples to show that neither cocompact nor locally cocompact is equivalent to property L.","abstract_has_math":false,"creators":["Francis, Gerald L."],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1973,"date_issued":"1973","date_published":"1973","updated_at":"2026-07-22T22:19:51Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/74672","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Francis, Gerald L."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-01-30T21:23:40Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-01-30T21:23:40Z"]},{"key":"dc:date.issued","label":"Date","values":["1973"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/74672"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this paper we study the concept of cotopology in the areas of cocon·tinuous functions and cocompact spaces. Initially we investigate and provide needed results concerning closed bases for a topological space. We then study cocontinuous functions by relating them to various other weaker forms of continuous functions, namely c-continuous, almost continuous and weakly continuous. We show that if (Y,U) is locally compact T₂, then f:(X,T)-->(Y,U) is cocontinuous if and only if f⁻¹(0) ε T for every 0 ε U such that (Y - 0) is compact. We note that every almost continuous function is cocontinuous, and we provide conditions under which a weakly continuous function is cocontinuous. We also show that a cocontinuous function from a saturated space to a regular space is continuous. In the area of cocompact spaces we first provide a partial answer to a question of J. M. Aarts as to when the union of cocompact subsets of a space is cocompact. We show that the union of a closed cocompact subset and a closed compact subset is cocompact. We then introduce the properties, locally cocompact and somewhere cocompact, and relate them to property L which was introduced by R. McCoy. We show that every somewhere cocompact regular space has property L, and that every locally cocompact regular space has property L locally. We provide examples to show that neither cocompact nor locally cocompact is equivalent to property L."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Properties of cocontinuous functions and cocompact spaces"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Francis, Gerald L."],"dc:date.accessioned":["2017-01-30T21:23:40Z"],"dc:date.available":["2017-01-30T21:23:40Z"],"dc:date.issued":["1973"],"dc:description.abstract":["In this paper we study the concept of cotopology in the areas of cocon·tinuous functions and cocompact spaces. Initially we investigate and provide needed results concerning closed bases for a topological space. We then study cocontinuous functions by relating them to various other weaker forms of continuous functions, namely c-continuous, almost continuous and weakly continuous. We show that if (Y,U) is locally compact T₂, then f:(X,T)-->(Y,U) is cocontinuous if and only if f⁻¹(0) ε T for every 0 ε U such that (Y - 0) is compact. We note that every almost continuous function is cocontinuous, and we provide conditions under which a weakly continuous function is cocontinuous. We also show that a cocontinuous function from a saturated space to a regular space is continuous. In the area of cocompact spaces we first provide a partial answer to a question of J. M. Aarts as to when the union of cocompact subsets of a space is cocompact. We show that the union of a closed cocompact subset and a closed compact subset is cocompact. We then introduce the properties, locally cocompact and somewhere cocompact, and relate them to property L which was introduced by R. McCoy. We show that every somewhere cocompact regular space has property L, and that every locally cocompact regular space has property L locally. We provide examples to show that neither cocompact nor locally cocompact is equivalent to property L."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/74672"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Properties of cocontinuous functions and cocompact spaces"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:51Z"}