{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/38597"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/38597","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"The fine topology and other topologies on C(X,Y)","abstract":"\"The Fine Topology\" C(X,Y) where (Y,d) is a metric space is referred to, in an exercise in [14], as the topology generated by basic open neighborhoods of the form B(f,E) = {g: d(f(x),g(x)) < E(x)} where E is a positive continuous real valued function. So in the fine topology, a function g is close to f if g(x) is continuously close to f(x); whereas in the uniform topology, g(x) must be uniformly close to f(x), that is, within a constant distance of f(x). So the fine topology is an obvious refinement of the uniform topology. This topology has not been extensively studied before, and it is the purpose of this paper to see how the fine topology fits in with the lattice of other well studied topologies on C(X,Y), and to study some properties of this topology in itself. Furthermore, other results on these well studied topologies will-be examined and compared with the fine topology.","abstract_html":"&quot;The Fine Topology&quot; C(X,Y) where (Y,d) is a metric space is referred to, in an exercise in [14], as the topology generated by basic open neighborhoods of the form B(f,E) = {g: d(f(x),g(x)) &lt; E(x)} where E is a positive continuous real valued function. So in the fine topology, a function g is close to f if g(x) is continuously close to f(x); whereas in the uniform topology, g(x) must be uniformly close to f(x), that is, within a constant distance of f(x). So the fine topology is an obvious refinement of the uniform topology. This topology has not been extensively studied before, and it is the purpose of this paper to see how the fine topology fits in with the lattice of other well studied topologies on C(X,Y), and to study some properties of this topology in itself. Furthermore, other results on these well studied topologies will-be examined and compared with the fine topology.","abstract_has_math":false,"creators":["Eklund, Anthony D."],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["McCoy, Robert A."],"committee_members":["Aull, Charles E.","Greenberg, William","Johnson, Lee W.","Parry, Charles J."],"year":1978,"date_issued":"1978-05-05","date_published":"1978-05-05","updated_at":"2026-07-22T22:19:03Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-06092012-141053"],"render_values":[{"text":"etd-06092012-141053","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/38597","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["McCoy, Robert A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Aull, Charles E.","Greenberg, William","Johnson, Lee W.","Parry, Charles J."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Eklund, Anthony D."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:15:00Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:15:00Z","2012-06-09"]},{"key":"dc:date.issued","label":"Date","values":["1978-05-05"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"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. 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So in the fine topology, a function g is close to f if g(x) is continuously close to f(x); whereas in the uniform topology, g(x) must be uniformly close to f(x), that is, within a constant distance of f(x). So the fine topology is an obvious refinement of the uniform topology. This topology has not been extensively studied before, and it is the purpose of this paper to see how the fine topology fits in with the lattice of other well studied topologies on C(X,Y), and to study some properties of this topology in itself. Furthermore, other results on these well studied topologies will-be examined and compared with the fine topology."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["The fine topology and other topologies on C(X,Y)"]}]}],"canonical_facts":{"dc:contributor.committeechair":["McCoy, Robert A."],"dc:contributor.committeemember":["Aull, Charles E.","Greenberg, William","Johnson, Lee W.","Parry, Charles J."],"dc:contributor.department":["Mathematics"],"dc:creator":["Eklund, Anthony D."],"dc:date.accessioned":["2014-03-14T21:15:00Z"],"dc:date.available":["2014-03-14T21:15:00Z","2012-06-09"],"dc:date.issued":["1978-05-05"],"dc:description.abstract":["\"The Fine Topology\" C(X,Y) where (Y,d) is a metric space is referred to, in an exercise in [14], as the topology generated by basic open neighborhoods of the form B(f,E) = {g: d(f(x),g(x)) < E(x)} where E is a positive continuous real valued function. So in the fine topology, a function g is close to f if g(x) is continuously close to f(x); whereas in the uniform topology, g(x) must be uniformly close to f(x), that is, within a constant distance of f(x). So the fine topology is an obvious refinement of the uniform topology. This topology has not been extensively studied before, and it is the purpose of this paper to see how the fine topology fits in with the lattice of other well studied topologies on C(X,Y), and to study some properties of this topology in itself. Furthermore, other results on these well studied topologies will-be examined and compared with the fine topology."],"dc:description.degree":["Ph. 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