Topological Algebraic Structure in the Density Topology and on Souslin Lines

摘要

This research investigates which topological algebraic structures can exist on two types of topological spaces: the real line R with the density topology; and any linearly ordered topological space (LOTS) satisfying the countable chain condition (CCC) that is not separable (i.e. any Souslin Line). Some surprising results are established in the density topology when considering the common group operations on R. Indeed, this research shows that addition and multiplication are not topological group operations in this space. These theorems are then generalized to show that there are no topological group operations on R with the density topology. The case of cancellative topological semigroups, however, is left as an open question.On the other hand, the conditions of existence of topological algebraic structures on Souslin lines is rather completely determined by this work. The main results in this space are that paratopological groups do not exist on any Souslin line, but cancellative topological semigroups do exist. The research on this space culminates with the construction of a cancellative topological semigroup on a Souslin line.