The extension problem is one of the fundamental problems of topology; if TX and Y are topological spaces and A is a closed subset of X and f:A —> Y is a continuous function, under what conditions is there »a continuous function fzx + Y such‘ that fIA = f ? Two key theorems of topology deal with this problem. They are the Tietze Extension A Theorem and the Brouwer No—Retraction Theorem. The first states that 1‘:-X is a normal Hausdorff space and Y is the real line then any continuous function f from a closed subset A of X into Y can be extended to a continuous function from X into Y . The second states that if X is the (n +1) -ball and Y and A are both the n-sphere bounding X in En+1 then the identity mapping from A to Y cannot be extended to a continuous function from X to Y.
","abstract_html":"<p>The extension problem is one of the fundamental problems of topology; if TX and Y are topological spaces and A is a closed subset of X and f:A —> Y is a continuous function, under what conditions is there »a continuous function fzx + Y such‘ that fIA = f ? Two key theorems of topology deal with this problem. They are the Tietze Extension A Theorem and the Brouwer No—Retraction Theorem. The first states that 1‘:-X is a normal Hausdorff space and Y is the real line then any continuous function f from a closed subset A of X into Y can be extended to a continuous function from X into Y . The second states that if X is the (n +1) -ball and Y and A are both the n-sphere bounding X in En+1 then the identity mapping from A to Y cannot be extended to a continuous function from X to Y.</p>","abstract_has_math":false,"creators":["Pixley, Carl Preston"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":["Prabir Roy","Louis F. McAuley","Harry W. Berkowitz"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1972,"date_issued":"1972-01-01T08:00:00Z","date_published":"1972-01-01T08:00:00Z","updated_at":"2026-07-24T01:10:28Z","subjects":["Topological spaces","Topology","Selection theory","Infinite dimensional spaces","Continuous single-valued approximations","Semi-continuous multivalued mappings"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://orb.binghamton.edu/dissertation_and_theses/181","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Prabir Roy","Louis F. McAuley","Harry W. Berkowitz"]},{"key":"dc:creator","label":"Author","values":["Pixley, Carl Preston"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Topological spaces","Topology","Selection theory","Infinite dimensional spaces","Continuous single-valued approximations","Semi-continuous multivalued mappings"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://orb.binghamton.edu/dissertation_and_theses/181"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The extension problem is one of the fundamental problems of topology; if TX and Y are topological spaces and A is a closed subset of X and f:A —> Y is a continuous function, under what conditions is there »a continuous function fzx + Y such‘ that fIA = f ? Two key theorems of topology deal with this problem. They are the Tietze Extension A Theorem and the Brouwer No—Retraction Theorem. The first states that 1‘:-X is a normal Hausdorff space and Y is the real line then any continuous function f from a closed subset A of X into Y can be extended to a continuous function from X into Y . The second states that if X is the (n +1) -ball and Y and A are both the n-sphere bounding X in En+1 then the identity mapping from A to Y cannot be extended to a continuous function from X to Y.
"]},{"key":"dc:title","label":"Title","values":["Selection theory for infinite dimensional spaces and continuous single-valued approximations to upper semi-continuous multivalued mappings"]}]}],"canonical_facts":{"dc:contributor":["Prabir Roy","Louis F. McAuley","Harry W. Berkowitz"],"dc:creator":["Pixley, Carl Preston"],"dc:description.abstract":["The extension problem is one of the fundamental problems of topology; if TX and Y are topological spaces and A is a closed subset of X and f:A —> Y is a continuous function, under what conditions is there »a continuous function fzx + Y such‘ that fIA = f ? Two key theorems of topology deal with this problem. They are the Tietze Extension A Theorem and the Brouwer No—Retraction Theorem. The first states that 1‘:-X is a normal Hausdorff space and Y is the real line then any continuous function f from a closed subset A of X into Y can be extended to a continuous function from X into Y . The second states that if X is the (n +1) -ball and Y and A are both the n-sphere bounding X in En+1 then the identity mapping from A to Y cannot be extended to a continuous function from X to Y.
"],"dc:identifier":["https://orb.binghamton.edu/dissertation_and_theses/181"],"dc:subject":["Topological spaces","Topology","Selection theory","Infinite dimensional spaces","Continuous single-valued approximations","Semi-continuous multivalued mappings"],"dc:title":["Selection theory for infinite dimensional spaces and continuous single-valued approximations to upper semi-continuous multivalued mappings"],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T01:10:28Z"}