The purpose of this thesis is to show the use of topology in mathematical logic. It is well known, through the work of H. Rasiowa and R. Sikorski, that Intuitionistic first order predicate calculus with equality has a very natural interpretation in terms of the lattice of open sets of a topological space. The problem was to show that the usual Intuitionistic second order arithmetic gives such a topological model. In the first two chapters we give the topology required for later work and a brief discussion on Intuitionistic mathematics. In the topology section the emphasis is on the Baire Space as this is the most natural space in which to interpret second order arithmetic. The discussion on Intuitionistic mathematics gives an Intuitionistic viewpoint but concentrates on giving, where possible, Classical justification for Intuitionistic assumptions and theorems. Chapter Three is a brief description of Ra owa's and Sikorski' s work 'showing that Intuitionistic first order predicate calculus with equality has topological models. Chapter Four gives a model for Intuitionistic second order arithmetic which is due to J.R. Moschovakis and is an adaption of topological models for Intuitionistic analysis given by D. Scott. Chapter Five shows a second use of topology in mathematical logic. It deals with the topological approach of forcing due to G. Takeuti and C. Ryll-Nardzewski. The central idea is that by using the Baire Category Theorem we can show the existence of generic models without giving a constructive proof. Chapter One (except 1.2.9 and 1.3.7) and the proof of 2.5.7 are taken from lectures given by R.A. Bull to a 1976 Honours III Logic class. The details of Chapter Five were worked out with the help of R.A. Bull.