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Open Educational Resources (OER) in the Discipline of Mathematics (July 2018): Geometry and Topology

By Andrew Misseldine

Geometry and Topology

Unfortunately, few high quality OER textbooks exist in the realms of geometry and topology for the undergraduate curriculum. For geometry, this seems mainly due to the fact that institutions vary widely in their curricula for undergraduate modern geometry courses. The leader in this category is Geometry with an Introduction to Cosmic Topology by Michael P. Hitchman. Similar to the works by Beezer and Judson, Hitchman’s text is cleanly designed and polished; it is also available in several formats, including pdf, online, and print. The textbook’s website also offers useful supplementary materials and online learning software. Although the textbook is well written and organized, its table of contents seems to focus more on niche subject matter rather than topics more generally taught in undergraduate geometry. It emphasizes topological aspects of geometry and not at all on the axiomatics of modern geometry like Greenberg’s standard-setting (and traditionally published) text, Euclidean and Non-Euclidean Geometries. In fact, no OER geometry textbook presents a comprehensive study of geometry through Hilbert’s axioms. Other notable college geometry textbooks include Euclidean Plane and Its Relatives by Anton Petrunin and Elementary College Geometry by Henry Africk.

In topology, there is a similar dearth of high quality topology texts for undergraduates. (Interestingly, this is not the case for graduate topology textbooks, with special mention given to Allen Hatcher’s excellent Algebraic Topology). This lack is all the more inexplicable because point-set topology has a much more standard curriculum than does geometry. Consider this an appeal to the mathematical community: more undergraduate topology OER are sorely needed!