11865

10890

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1395

هشت، [78]ص.: مصور

ص.ع. به فارسي و انگليسي

محمدرضا كوشش

واژه نامه

مجيد فخار، رامين جوادي

1395/09/02

كتابنامه

رياضي

ID10890

Obtainable Sizes of Topologies on Finite Sets Narges Rezaei narges rezaei@math iut ac ir 2016 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Mohammad Reza Koushesh koushesh@cc iut ac ir Second Supervisor Dr Gholam Reza Omidi romidi@cc iut ac ir 2010 MSC 54A25 05B25 Keywords Finite topology Poset Order ideal Integer sequence Labeled topology Unlabeledtopology AbstractThis thesis is based on the works done by K ri Ragnarsson and Bridget Eileen Tenner 32 andGunnar Brinkmann and Brendan D McKay 3 There has been a fundamental problem to determine the number of di erent non homeomorphic topologies de ned on a nite set of n points This number has been determined for small numbers byenumeration however the general problem is very di cult and has been so far remained open Thisnumber has been estimated asymptotically indeed it is known that the number of topologies on npoints is asymptotically the same as the number of T0 topologies on n points for which asymptoticbounds exists The general problem of determination of the number of topologies on n points may be reduced to theproblem of nding the number of topologies on n points with k open sets We denote this numberby T n k As for the case of the general problem this problem is also open though several partialresults exist Our study in this thesis is divided into two parts In the rst part we study the smallest possible number of points in a topological space having kopen sets Equivalently this is to study the smallest possible algorithms for constructing a topologywith a prescribed size We show that this number has a logarithmic upper bound This is done byde ning a preorder relation on the topological space We then construct a one to one correspondence

محمدرضا كوشش

مجيد فخار، رامين جوادي