7. Topological Ethics

Yesterday I was thinking about car insurance. You see, we might be moving to Montreal in a few weeks, and we have discussed the possibility of maintaining the lease on our apartment here so as to keep our address available to be used for things like car registration and insurance. I have certain misgivings about this arrangement- because generally they want to give you a quote based on "where your car is garaged at night" or some such phrase. (We have checked, and our car insurance company doesn't provide coverage for residents of Canada- although we are covered when visiting Canada.) I believe in being honest and Following the Rules in life in almost every area- for example, I've paid hundreds of dollars in taxes on my private tutoring money, when most everyone I know told me "You're crazy- the government has no record of your private tutoring- just don't report it." But there are a few things, like speeding and listening to copies of copywrited CD's, that I let myself get away with. I can be a champion Rationalizer- a real danger for conscientious living, indeed- but I had a mathematically euphoric moment yesterday when I realized that

Okay, back up for a second. Back to the insurance. This is what I've been thinking: If we were only going to visit Montreal for one week, we could keep PA insurance, no problems. Well, how about a month? Even if we rented an apartment in Montreal for a month, that wouldn't really matter. It would still be like a Visit. Well, how about a month and a day? Or two months? You see where I'm going with this. How about if we were itinerant mathematicians (or musicians, or ventriloquists... you get the idea.) What would be our "address" for insurance purposes, then?

So what I realized in my mathematically euphoric moment is that my rationalization scheme (in this particular case, at least) is really what Nate (an ardent amateur philophilosophist*) would call a "line-drawing fallacy," but what I would think of as a- for lack of a better more professional term- DEFORMATION EQUIVALENCE CLASS ANALYSIS of ETHICAL DILEMMAS.

What does this have to do with Topology, you may wonder?

(*This is arguably the most redundant phrase ever to appear on the Nebiverse. I basically just said that he's a lovingly loving lover of knowledge.)

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Definition: Let X be a set. A Topology T on the set X is a collection of subsets of X that satisfies the following properties:
1) The empty set is in T.
2) Any finite intersection of sets in T is again in T.
3) Any arbitrary union of sets in T is again in T. ("Arbitrary" here means it can be the union of finitely many or infinitely many sets.)

Definition: Let X be a set with a topology T. Any set that is in T is called an "open" set.
Definition: A set is "closed" if its complement is "open."

As a good example, think of X as being the real numbers; the "standard" topology on the reals is defined as follows: Let S be a set of real numbers. S is an "open" set if, for every x in S, there exists a number y greater than 0 such that the interval (x-y, x+y) is contained in S.

The empty set and the whole set are both open and closed.
Many sets are neither open nor closed.

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And from all this, mathematicians launch Topology. [The above was Page 1 of The Topology Textbook in Neb's Head, by the way. Did I get anything major wrong? It's late and I'm sleepy. Erratum should be reported to the management in the morning, with kindness and charity and coffee and some soft words, please. Thanks much. Also, a morning paper would be nice.] And the most popular Tourist Attraction in Topology is the quaint little island where it is determined that

A Coffee Cup And A Donut Are Really The Same Thing!!

because one can be Deformed continuously into the other.

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The car insurance company's Rule cannot be upheld held up, because it can be continuously deformed into Oblivion. A flat line. The Null Rule. Therefore, I am at Peace.

And now, very sleepy.

Good night!
Love, Neb