Topological groups

Abstract

Namen tega diplomskega dela je predstaviti pojem topološke grupe in dokazati nekaj temeljnih izrekov iz študija topoloških grup. Definirana je topološka grupa in opisane so njene osnovne lastnosti. Obravnavane so topološke podgrupe in kvocientni topološki prostori topoloških grup. Pokazano je, da za topološke grupe veljajo podobni trije izreki o topoloških izomorfizmih kot za grupe. Na topološko grupo sta vpeljani leva in desna uniformna struktura, glede na kateri je vsaka topološka grupa uniformni prostor. Na topološki grupi je nato skonstruirana levoinvariantna psevdometrika. Karakterizirana je metrizabilnost za Hausdorffove topološke grupe in dokazano je, da sta za topološke grupe povsem regularnost in separacijski aksiom $T_0$ ekvivalentna. Skonstruiran je primer povsem regularne topološke grupe, ki ni normalna. Za regularne topološke prostore so navedene karakterizacije parakompaktnosti. Dokazano je, da je vsaka lokalno kompaktna Hausdorffova topološka grupa parakompaktna in posledično normalna.The goal of this thesis is to present the concept of a topological group and to prove some fundamental theorems from the study of topological groups. We define a topological group and describe its basic properties. We look at topological subgroups and quotient topological spaces of topological groups. We show that for topological groups three topological isomorphism theorems hold which are similar to those for groups. We introduce left and right uniform structures on a topological group and then show that every topological space is also a uniform space. We then construct a left invariant pseudo-metric on a topological group. We characterize metrizability for Hausdorff topological groups and we prove that complete regularity and the $T_0$ separation axiom are equivalent for topological groups. We construct an example of a completely regular topological group which is not a normal topological space. For regular topological spaces we list different characterizations of paracompactness. We then prove that every locally compact Hausdorff topological group is paracompact, and hence a normal topological space