in article <email@example.com>, dave rusin <firstname.lastname@example.org> wrote:
|In article <email@example.com>, |James Dolan <firstname.lastname@example.org> wrote: |>in article |><waderameyxiii-E2E22B.email@example.com>, the |>world wide wade <firstname.lastname@example.org> wrote: | |>|And if you haven't seen metric |>|spaces, the axioms for point-set topology will appear like some |>|arbitrary abstract nonsense from another planet. |> |>grossly false; | |OK, I have a bright student uncorrupted by point-set topology before |me. I present the definition of a topology; the collection of open |sets is closed under _finite_ intersections but _arbitrary_ |unions. The student asks why on earth one would take such an |asymmetrical set of axioms. Your answer? | |A bit later we have to decide what the morphisms of the category are. |I hope you don't consider it a corrupting previous specialization |that the student has already encountered homomorphisms of groups and |rings. So now we have topological spaces: sets and preferred |subsets. We define the topological maps to be: functions between the |sets that, um, do what?! you define the appropriate morphisms so that |the INVERSE images of the preferred sets in Y are preferred sets in |X? Why on earth would you do that? Your answer? | |I'm all about not dwelling on minutiae that hide rather than |highlight What's Really Going On. That's cool. On the other hand, I |can't imagine hiding the origins of the definitions, the conjectures |that prompted the theorems, etc. I don't know about you but I'm not |interested in preparing desert-island mathematicians who have |discovered all the consequences of a set of axioms that no one cares |about in the least. My students all want to be part of a larger |culture -- at least a mathematical one -- and they want to know why |the goofy axioms I present might possibly be connected to anything |else at all. So I always spend some time on historically-important |special cases. | |Guess I'm just doing it wrong then.
i have to question your reading comprehension if you think that your questions here are somehow responsive to something i wrote. however if we disregard the issue of what inspired your questions and just consider them as dropped out of thin air for no particular reason then i don't mind spending a minute or two answering a couple of them.
first consider what happens if you omit the cardinality restriction in the definition of topology. namely, there's an elegant lemma (with proof probably shorter than the statement of the lemma) that such topologies on a set are precisely equivalent to pre-orders (which are structures of a lower level of complexity, more directly accessible to the intuition), and that a map is pre-order-preserving precisely in case the inverse images of open sets are open.
it's then obvious that topologies in general are ideal refinements of pre-orders, and that continuous maps in general are ideal refinements of pre-order-preserving maps, and this provides the appropriate geometric intuition for understanding topological spaces and continuous maps as tools for studying "cohesion" in a context where pre-orders are refined more and more finely without limit.
there's a lot more that can be said to help students understand the details as well as the broad currents of ideas here and anyone who'd like to pay me to say more of it is welcome to make an offer.