Stephen Barr’s “Experiments in Topology” (originally published in 1964, reprinted in 1989 by Dover) is extraordinary because it treats a sophisticated mathematical subject with accessible language that can be understood by motivated junior high students. It is extraordinary because its wonderful and copious figures are remarkably clear and elucidating. It is extraordinary because it captures the flavor, depth, and breadth of a very subtle subject with carefully written passages that boil down significant complications into understandable overviews. It is extraordinary because he gives a concrete enough treatment that the attentive reader can learn something substantial of the subject while the mysteries that are both alluded to and implicit may drive the curious reader to explore its nooks and crannies. It is extraordinary because its learning-by-doing (experimental / exploring) style is infectious and the reader may be emboldened to ask their own questions and attempt their own experiments. It is extraordinary because its effective survey of key ideas in the major branches of topology make it a useful reference. It is extraordinary because it can reward the casual reader with some basic guideposts for apprehending an advanced subject while the serious student who builds all its models and tries to understand their integrated significance can extract many deeper insights from the text.

I prefer this latter, in-depth, approach and organized, through Math Counts, a mathematics group in Philadelphia, 11 deep explorations on topics from the book each with a group of 6-10 mathematically-oriented colleagues. We thoroughly enjoyed the book and our explorations, although after 11 months we wanted to move on to other ideas.

The book uses homeomorphism as its first principle for exploring topology. Barr gives several definitions, but I found the characterization on page 5 to be the most helpful: “Any distortion is allowed provided the end result is connected in the same way as the original.” This exemplifies the kind of careful but informal style of the book.

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