The aim of this thesis is to discover some of the concept images held by students concerning metric spaces. I was specifically interested in the concept of an open set in a metric space. This work also addresses the topics of distance and open balls, which are both essential when considering open sets. Also, I sought to investigate the relationships between students’ concept images and the concept definition, and the uses students make of their concept images and the concept definition when working on problems. Furthermore, this thesis explores some of the attitudes of students to mathematical definitions, and their understanding of the role definitions play in mathematics. A set of questions was designed in order to gain insight into students’ understanding of the concept of open set, and their opinions on mathematical definitions. The questions were used in a written questionnaire, which was completed in class by 16 students who were enrolled in a metric spaces course and in individual interviews with eleven students. From the results of the study I was able to categorise the concept images of an open set into five categories: those based on the definition; those based on boundary points; those based on a union of open balls; those based on open sets in Euclidean space; and those based on visualisation. I also categorised the concept images of distance into four categories: concept image based on the measurement of similarity; those based on comparison/difference between points; those based on physical distance; and those based on the formal definition. I was able to uncover some influences on the formation of concept images, and some misconceptions related the concept of an open set. The study also discovered that these students generally understand the role of definitions in advanced mathematics; however they often consider and use their conceptions in the place of definitions.