One nice example (from topology) is Tychonoff's Theorem (that a product of compact spaces is compact). No many matter how many times I see it, I find the classic proof based on the (Alexandre Subbase Lemma) difficult and opaque. On the other hand if one first develops the theory of nets (aka Moore-Smith Convergence), not only is that a powerful tool for all sorts of other purposes, but its development is a natural and intuitive generalization of sequences, and the place where Zorn's Lemma enters (the proof that any net has a universal subnet) is much clearer than in the proof of the subbase lemma. And of course once one has universal nets, the proof of Tychonoff is the obvious generalization of the trivial proof that a finite product of sequentially compact spaces is sequentially compact.
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One nice example (from topology) is Tychonoff's Theorem (that a product of compact spaces is compact). No many how many times I see it, I find the classic proof based on the (Alexandre Subbase Lemma) difficult and opaque. On the other hand if one first develops the theory of nets (aka Moore-Smith Convergence), not only is that a powerful tool for all sorts of other purposes, but its development is a natural and intuitive generalization of sequences, and the place where Zorn's Lemma enters (the proof that any net has a universal subnet) is much clearer than in the proof of the subbase lemma. And of course once one has universal nets, the proof of Tychonoff is the obvious generalization of the trivial proof that a finite product of sequentially compact spaces is sequentially compact. |
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