Given two topologies, T_{1} and T_{2} on the same set X , the intersection topology with respect to T_{1} and T_{2} is the topology with basis {U_{1} ∩ U_{2} : U_{1} Є T_{1}, U_{2} Є T_{2}} Equivalently, T is the join of T_{1} and T_{2} in the lattice of topologies on the set X . This thesis is concerned with analysing some particular classes of intersection topologies, and also with making some more general remarks about the technique. Reed was the first to study intersection topologies in these terms, and he made an extensive investigation of intersection topologies on a subset of the reals of cardinality N_{1}, where the topologies under consideration are the inherited realline topology and the topology induced by an ω_{1}type ordering of the set. We consider the same underlying set, and describe the properties of the intersection topology with respect to the inherited Sorgenfrey line topology and an ω_{1}type order topology, demonstrating that, whilst most of the properties possessed by Reed's class are shared by ours, the two classes are strictly disjoint. A useful characterisation of the intersection topology is as the diagonal of the product of the two topologies under consideration. We use this to prove some general properties about intersection topologies, and also to show that the intersection topology with respect to a first countable, hereditarily separable space and an ω_{1}type order topology can never be locally compact. Results about the real line and Sorgenfrey line intersections with ω_{1} use various properties of the two lines. We demonstrate that most of the basic properties of the intersection topology require only the hereditary separability of R, and give examples to show that 'hereditary' is essential here. We also show that results about normality, ω_{1}compactness and the property of being perfect, all of which are settheoretic in the classes of realω_{1} and Sorgenfreyω_{1} intersection topologies, can be shown to generalise to the class of intersection topologies with respect to separable generalised ordered spaces and ω_{1}.
