Show simple item record

dc.contributor.advisorVobach, Arnold R.
dc.contributor.authorReynolds, Donald Fainen_US
dc.date.accessioned2019-10-11T15:11:02Z
dc.date.available2019-10-11T15:11:02Z
dc.date.created1970en_US
dc.date.issued1970en_US
dc.identifieraleph-255076en_US
dc.identifier.urihttps://repository.tcu.edu/handle/116099117/33813
dc.description.abstractLet (X, T) be a topological space and let A be a subset of X. The topology generated by T and A is defined to be the simple extension of the topology T to the set A. Sone elementary properties of simple extensions of topologies are examined. Necessary and sufficient conditions are given for the preservation of various hereditary and weakly hereditary topological properties under a simple extension of the topology. It is shown that connectedness is preserved if A is a dense set. The use of simple extensions in the construction of certain types of counterexamples is discussed. The more general concepts of finite and infinite extensions of topologies are defined to be the topology generated by T and a given collection {A_alpha | alpha in Gamma} of subsets of x. Necessary and sufficient conditions are given for the preservation of various topological properties under finite extensions of topologies. It is shown that every open set in an infinite extension topology can be expressed as a union of sets, each of which is open in some related finite extension topology. By placing various restrictions on the collection {A_alpha | alpha in Gamma}, it is shown that regularity, complete regularity, paracompactness, metrizability, normality and connectedness are preserved under infinite extensions of topologies.
dc.format.extentiv, 42 leaves, bounden_US
dc.format.mediumFormat: Printen_US
dc.language.isoengen_US
dc.relation.ispartofTexas Christian University dissertationen_US
dc.relation.ispartofAS38.R49en_US
dc.subject.lcshTopologyen_US
dc.titlePreservation of topological properties under extensions of topologiesen_US
dc.typeTexten_US
etd.degree.departmentDepartment of Mathematics
etd.degree.levelDoctoral
local.collegeCollege of Science and Engineering
local.departmentMathematics
local.academicunitDepartment of Mathematics
dc.type.genreDissertation
local.subjectareaMathematics
dc.identifier.callnumberMain Stacks: AS38 .R49 (Regular Loan)
dc.identifier.callnumberSpecial Collections: AS38 .R49 (Non-Circulating)
etd.degree.nameDoctor of Philosophy
etd.degree.grantorTexas Christian University


Files in this item

FilesSizeFormatView

There are no files associated with this item.

This item appears in the following Collection(s)

Show simple item record