{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/76130"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/76130","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"On M-spaces and M*-spaces","abstract":"In this thesis we investigate the properties of M-spaces and M*-spaces, which are generalized metric spaces. Chapter II is devoted to preliminary results, and in Chapter III we prove the characterization for M-spaces theorem of K. Morita [12]. This theorem states that a space X is an M-space if and only if there exists a quasi-perfect map from X onto a metrizable space T. Chapter IV is concerned with the relationships between M-spaces and M*-spaces. We first prove an M-space is an expandable, M*'-space and then show that every normal, expandable, M*-space is an M-space. Using Katetov's Theorem, we show that in a collectionwise normal space, X is an M-space if and only if it is an M*-space. We conclude by generalizing this to the following. In a normal space X, X is an M-space if and only if it is an M*-space. Chapter V is concerned with the study of M-spaces and M*-spaces under quasi-perfect maps. We also prove the Closed Subspace Theorem for M-spaces and M*-spaces and establish the Locally Finite Sum Theorem for M-spaces and M*-spaces. In Chapter VI, we give an example of a T₂, locally compact M-space X, which is not normal and therefore not metrizable. We also give an example of a T₂, locally compact M*-space Y, which is not an M-space, but is however the image of X under a quasi-perfect mapping.","abstract_html":"In this thesis we investigate the properties of M-spaces and M*-spaces, which are generalized metric spaces. Chapter II is devoted to preliminary results, and in Chapter III we prove the characterization for M-spaces theorem of K. Morita [12]. This theorem states that a space X is an M-space if and only if there exists a quasi-perfect map from X onto a metrizable space T. Chapter IV is concerned with the relationships between M-spaces and M*-spaces. We first prove an M-space is an expandable, M*'-space and then show that every normal, expandable, M*-space is an M-space. Using Katetov's Theorem, we show that in a collectionwise normal space, X is an M-space if and only if it is an M*-space. We conclude by generalizing this to the following. In a normal space X, X is an M-space if and only if it is an M*-space. Chapter V is concerned with the study of M-spaces and M*-spaces under quasi-perfect maps. We also prove the Closed Subspace Theorem for M-spaces and M*-spaces and establish the Locally Finite Sum Theorem for M-spaces and M*-spaces. In Chapter VI, we give an example of a T₂, locally compact M-space X, which is not normal and therefore not metrizable. We also give an example of a T₂, locally compact M*-space Y, which is not an M-space, but is however the image of X under a quasi-perfect mapping.","abstract_has_math":false,"creators":["Nuckols, Thomas Ryland"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1970,"date_issued":"1970","date_published":"1970","updated_at":"2026-07-22T22:19:21Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/76130","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":["Nuckols, Thomas Ryland"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-03-10T18:28:50Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-03-10T18:28:50Z"]},{"key":"dc:date.issued","label":"Date","values":["1970"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"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":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"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"]},{"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/10919/76130"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis we investigate the properties of M-spaces and M*-spaces, which are generalized metric spaces. Chapter II is devoted to preliminary results, and in Chapter III we prove the characterization for M-spaces theorem of K. Morita [12]. This theorem states that a space X is an M-space if and only if there exists a quasi-perfect map from X onto a metrizable space T. Chapter IV is concerned with the relationships between M-spaces and M*-spaces. We first prove an M-space is an expandable, M*'-space and then show that every normal, expandable, M*-space is an M-space. Using Katetov's Theorem, we show that in a collectionwise normal space, X is an M-space if and only if it is an M*-space. We conclude by generalizing this to the following. In a normal space X, X is an M-space if and only if it is an M*-space. Chapter V is concerned with the study of M-spaces and M*-spaces under quasi-perfect maps. We also prove the Closed Subspace Theorem for M-spaces and M*-spaces and establish the Locally Finite Sum Theorem for M-spaces and M*-spaces. In Chapter VI, we give an example of a T₂, locally compact M-space X, which is not normal and therefore not metrizable. We also give an example of a T₂, locally compact M*-space Y, which is not an M-space, but is however the image of X under a quasi-perfect mapping."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["On M-spaces and M*-spaces"]}]}],"canonical_facts":{"dc:contributor.department":["Mathematics"],"dc:creator":["Nuckols, Thomas Ryland"],"dc:date.accessioned":["2017-03-10T18:28:50Z"],"dc:date.available":["2017-03-10T18:28:50Z"],"dc:date.issued":["1970"],"dc:description.abstract":["In this thesis we investigate the properties of M-spaces and M*-spaces, which are generalized metric spaces. Chapter II is devoted to preliminary results, and in Chapter III we prove the characterization for M-spaces theorem of K. Morita [12]. This theorem states that a space X is an M-space if and only if there exists a quasi-perfect map from X onto a metrizable space T. Chapter IV is concerned with the relationships between M-spaces and M*-spaces. We first prove an M-space is an expandable, M*'-space and then show that every normal, expandable, M*-space is an M-space. Using Katetov's Theorem, we show that in a collectionwise normal space, X is an M-space if and only if it is an M*-space. We conclude by generalizing this to the following. In a normal space X, X is an M-space if and only if it is an M*-space. Chapter V is concerned with the study of M-spaces and M*-spaces under quasi-perfect maps. We also prove the Closed Subspace Theorem for M-spaces and M*-spaces and establish the Locally Finite Sum Theorem for M-spaces and M*-spaces. In Chapter VI, we give an example of a T₂, locally compact M-space X, which is not normal and therefore not metrizable. We also give an example of a T₂, locally compact M*-space Y, which is not an M-space, but is however the image of X under a quasi-perfect mapping."],"dc:description.degree":["Master of Science"],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/76130"],"dc:language.iso":["en"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["On M-spaces and M*-spaces"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:21Z"}