Timeline for Why is a topology made up of 'open' sets?
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| Mar 30, 2015 at 5:22 | comment | added | Martin Brandenburg | This great answer reminds me of several "unbiased" or "monadic" definitions of structures such as monoids or monoidal categories. The definition of a monoid "should" really introduce $n$-fold products for $n \in \mathbb{N}$, not just $0$-fold and $2$-fold products. Curiously, the monoid axioms are quite easy to state for $n$-fold products; this is even more true for the coherence axioms in the definition of a monoidal category via $n$-fold tensor products. | |
| May 8, 2012 at 2:46 | comment | added | temp | I really like this answer. Perhaps part of the so-called "mathematical maturity" is being able to recover the full package from the minimalist's definition. I'm not always able to do so. It took me a long time to realize the metric tensor really gives you a "package". I wish someone can tell me what the package the minimal definition of a scheme (as a ringed space $(X,\mathcal{O}_X)$ that is locally isomorphic to the spectrum of a ring) gives... Although I feel like I can handle it freely right now, I certainly don't know how to explain it to someone who is new to algebraic geometry. | |
| Jun 27, 2011 at 14:15 | history | made wiki | Post Made Community Wiki | ||
| Jul 2, 2010 at 12:38 | comment | added | Jacques Carette | The difference between the minimalist and 'packaged' version is exactly what Bill Farmer and I have called axiomatic theories and high-level theories (respectively). A minimalist 'basis' is useful when trying to connect theories (theory interpretations, aka homomorphisms) because less has to be proven, but that makes a really poor "working environment". Any mathematician using a theory (be it topology or Riemannian geometry) will automatically want to use the high-level version. When trying to mechanize mathematics, such issues are no longer philosophical! | |
| Jul 2, 2010 at 5:09 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Jul 1, 2010 at 20:46 | history | edited | Terry Tao | CC BY-SA 2.5 |
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| Jul 1, 2010 at 20:40 | history | answered | Terry Tao | CC BY-SA 2.5 |