One might argue that (exotic) counterexamples fall into the first category. For example, when I took Munkres' course in topology it was organized around many very carefully chosen (counter)examples showing how various properties were related. This led me me to delve into many exotic spaces listed in Steen and Seebach's Counterexamples in Topology. While I'll probably never make use of any of those exotic counterexamples, it did help me to learn better how to employ the axiomatic method, e.g. to understand how to constuct examples showing that axioms are independent, to construct pertinent examples for theoretical signposts. summits, etc. One isn't necessarily expected to remember the examples but, rather, the methodology (e.g. various ideas of completion, compactification, ...)
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One might argue that (exotic) counterexamples fall into the first category. For example, when I took Munkres course in topology it was organized around many very carefully chosen (counter)examples showing how various properties were related. This led me me to delve into many exotic spaces listed in Steen and Seebach's Counterexamples in Topology. While I'll probably never make use of any of those exotic counterexamples, it did help me to learn better how to employ the axiomatic method, e.g. to understand how to constuct examples showing that axioms are independent, to construct pertinent examples for theoretical signposts. summits, etc. One isn't necessarily expected to remember the examples but, rather, the methodology (e.g. various ideas of completion, compactification, ...) |
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