{"id":{"repo_id":"south-carolina","oai_identifier":"oai:scholarcommons.sc.edu:etd-2597"},"canonical_url":"https://search.dev.ndltd.org/etd/south-carolina/oai:scholarcommons.sc.edu:etd-2597","repository":{"repo_id":"south-carolina","name":"University of South Carolina","base_url":"https://scholarcommons.sc.edu/do/oai/"},"display":{"title":"D-Spaces and L-Special Trees","abstract":"

This dissertation concerns D-spaces and set-theoretic trees. A topological space, X, is a D-space if for every neighbornet of the space there is a closed, discrete set from X whose images in the neighbornet are a cover for X. A set-theoretic tree is a poset where for any element the set of its predecessors is well-ordered.

In this dissertation it is shown that certain L-special trees are D-spaces and some of them are hereditarily so.In particular, let L=[0,1]α with α a countable ordinal be given the lexicographic order. For α < ω+1, the author shows that any L-special tree is hereditarily a D-space. For certain α with ω < α < ω1 the author shows that any L-special tree is a D-space. For the remaining countable ordinals α, the current progress is shown.

","abstract_html":"<p>This dissertation concerns D-spaces and set-theoretic trees. A topological space, X, is a D-space if for every neighbornet of the space there is a closed, discrete set from X whose images in the neighbornet are a cover for X. A set-theoretic tree is a poset where for any element the set of its predecessors is well-ordered.</p> <p>In this dissertation it is shown that certain L-special trees are D-spaces and some of them are hereditarily so.In particular, let L=[0,1]<super>&alpha;</super> with &alpha; a countable ordinal be given the lexicographic order. For &alpha; < &omega;+1, the author shows that any L-special tree is hereditarily a D-space. For certain &alpha; with &omega; < &alpha; < &omega;<sub>1</sub> the author shows that any L-special tree is a D-space. For the remaining countable ordinals &alpha;, the current progress is shown.</p>","abstract_has_math":false,"creators":["Gamel, Heather"],"institution":null,"degree_name":"Ph.D.","degree_level":"Campus Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Peter J Nyikos"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-01-01T08:00:00Z","date_published":"2011-01-01T08:00:00Z","updated_at":"2026-07-24T04:37:49Z","subjects":["Mathematics","Physical Sciences and Mathematics","Set Theoretic Topology"],"languages":[],"rights":["© 2011, Heather Gamel"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1596","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Peter J Nyikos"]},{"key":"dc:creator","label":"Author","values":["Gamel, Heather"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics","Set Theoretic Topology"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2011, Heather Gamel"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1596"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["

This dissertation concerns D-spaces and set-theoretic trees. A topological space, X, is a D-space if for every neighbornet of the space there is a closed, discrete set from X whose images in the neighbornet are a cover for X. A set-theoretic tree is a poset where for any element the set of its predecessors is well-ordered.

In this dissertation it is shown that certain L-special trees are D-spaces and some of them are hereditarily so.In particular, let L=[0,1]α with α a countable ordinal be given the lexicographic order. For α < ω+1, the author shows that any L-special tree is hereditarily a D-space. For certain α with ω < α < ω1 the author shows that any L-special tree is a D-space. For the remaining countable ordinals α, the current progress is shown.

"]},{"key":"dc:title","label":"Title","values":["D-Spaces and L-Special Trees"]}]}],"canonical_facts":{"dc:contributor":["Peter J Nyikos"],"dc:creator":["Gamel, Heather"],"dc:description.abstract":["

This dissertation concerns D-spaces and set-theoretic trees. A topological space, X, is a D-space if for every neighbornet of the space there is a closed, discrete set from X whose images in the neighbornet are a cover for X. A set-theoretic tree is a poset where for any element the set of its predecessors is well-ordered.

In this dissertation it is shown that certain L-special trees are D-spaces and some of them are hereditarily so.In particular, let L=[0,1]α with α a countable ordinal be given the lexicographic order. For α < ω+1, the author shows that any L-special tree is hereditarily a D-space. For certain α with ω < α < ω1 the author shows that any L-special tree is a D-space. For the remaining countable ordinals α, the current progress is shown.

"],"dc:identifier":["https://scholarcommons.sc.edu/etd/1596"],"dc:rights":["© 2011, Heather Gamel"],"dc:subject":["Mathematics","Physical Sciences and Mathematics","Set Theoretic Topology"],"dc:title":["D-Spaces and L-Special Trees"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T04:37:49Z"}