توپولوژي هاي القايي توسط متريك هاي معتبر درℓ_

چكيده انگليسي :

Topologies arising from metrics valued in abelian groups EBRAHIM RANJBARI e ranjbari@math iut ac ir January 11 2016 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mohammad Reza Koushesh koushesh@cc iut ac irAdvisor Dr Azam Etemad Dehkordy ae110mat@cc iut ac ir2011 MSC Primary 06F20 Secondary 54F05 54H12Keywords Lattice ordered groups generalized metric metric partial metric quasimetric valued in an group positive filter strong units weak units induced topology group topology Nachbin orderconvexity Abstract This thesis considers metrics valued in abelian groups and their induced topologies In addition to ametric into an group one needs a filter in the positive cone to determine which balls are neighborhoodsof their center As a key special case we discuss a topology on a lattice ordered abelian group from themetric dG and the filter of positives consisting of the weak units of G in the case of Rn this is theEuclidean topology Indeed a major reason for considering these natural distance structures is thatmany group topologies so arise and we use this in later sections to analyze them Next we show thatevery Tychonoff space can be derived from a generalized metric in the way described above and thatthe converse holds the following is the major result of this thesis Theorem a For each abelian group G Archimedean positive filter P on G and G metric d on aset X d P is a Tychonoff topology b Conversely given a Tychonoff topological space X there is an abelian group G an Archimedeanpositive filter P on G and a G metric d on X such that d P There are easy examples of metrics and positive filters that are not Archimedean but whose inducedtopology is Tychonoff For example using notation from Example 14 2 2 3 define d Q Q NA R by d x y x y 0 and let P VF We already know that VF is not Archimedean But if y Nr 1 x foreach r VF then x y 0 q 0 for each q 0 So x y which makes our topology T1 andhence Tychonoff We also show that there are many Nachbin convex topologies on an group which are not induced byany filter of the group For G C 0 1 the pointwise and Lp topologies p 1 are Nachbin convex group topologiesthat do not arise from a G metric dG and a positive filter on G group such as the fact that G P is the join of an upperMuch more topological structure on an abelian and a lower topology with respect to which and are continuous is brought out using metric likeproperties of the supremum function The appropriate metric like setting is also defined Partial metrics were introduced by the computer scientist Steve Matthews in 12 in the early 1990 s asa mechanism for studying how to gain increasingly precise but always partial knowledge about idealobjects For example to study gaining the knowledge that a number is in smaller and smaller intervals it is useful to study the poset of compact nonempty real intervals under the partial order While thereis no need to be familiar with the terminology there are three topologies involved the Scott in which

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توپولوژي هاي القايي توسط متريك هاي معتبر درℓ__ گروه هاي آبلي

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