Undergraduate Topology
A Working Textbook

This textbook offers an accessible, modern introduction at undergraduate level to an area known variously as general topology, point-set topology or analytic topology with a particular focus on helping students to build theory for themselves. It is the result of several years of the authors' combined university teaching experience stimulated by sustained interest in advanced mathematical thinking and learning, alongside established research careers in analytic topology. Point-set topology is a discipline that needs relatively little background knowledge, but sufficient determination to grasp ideas precisely and to argue with straight and careful logic. Research and long experience in undergraduate mathematics education suggests that an optimal way to learn such a subject is to teach it to yourself, pro-actively, by guided reading of brief skeleton notes and by doing your own spadework to fill in the details and to flesh out the examples. This text will facilitate such an approach for those learners who opt to do it this way and for those instructors who would like to encourage this so-called 'Moore approach', even for a modest segment of the teaching term or for part of the class. In reality, most students simply do not have the combination of time, background and motivation needed to implement such a plan fully. The accessibility, flexibility and completeness of this text enable it to be used equally effectively for more conventional instructor-led courses. Critically, it furnishes a rich variety of exercises and examples, many of which have specimen solutions, through which to gain in confidence and competence.

Aisling McCluskey graduated at Queen's University Belfast with a doctorate in Pure Mathematics (topology) in 1990 and subsequently was awarded a postdoctoral fellowship in Toronto in 1991. She was appointed to a permanent lectureship in Mathematics in the National University of Ireland (NUI), Galway, Ireland in January 1992. Since then, she has established a meaningful and rewarding academic career, maintaining an active research profile in the field whilst holding the teaching and learning of mathematics central to her academic endeavour. In recent years, her research interests have diversified into undergraduate mathematics education. She was awarded an institutional award in 2008 for excellence in teaching and then a national award in 2009 across all higher education institutions in Ireland. Brian McMaster studied at Queen's University Belfast, graduating with a PhD in 1972, and has served his alma mater department in various capacities including those of Adviser of Studies, Head of Research and Associate Director of Education. His publication profile covers over sixty refereed journal articles, mostly in the area of general topology but incorporating applications in disciplines as diverse as probabilistic metric spaces and decision support theory. He is presently formally retired but continues to deliver a full undergraduate teaching load on a voluntary basis, following his lifelong commitment to and passion for communicating mathematics to students. His teaching interests focus around analysis (real and complex) and set theory and their development into various fields especially that of analytic topology.