Hopefully Interesting

December 29, 2007

Topological Notation

Filed under: opinion — Pietro @ 7:19 am
Tags: ,

In general topology, one talks about open and closed sets a lot. A lot. So it seems a bit silly that there isn’t standard notation for that; it’s sort of like writing “equals” longhand throughout and entire semester of calculus. So I came up with the following simple symbols:

  • A \subseteq^\circ X (A is open in X);
  • A \subseteq^\bullet X (A is closed in X);
  • A^\circ (the interior of A);
  • A^\bullet (the closure of A).

They’ve been saving me a lot of time and thought since then, like notation’s supposed to. Witness A \subseteq^\circ B \subseteq^\circ X \Longrightarrow A \subseteq^\circ X . The closure symbol, in particular, has ended the ambiguity with \bar{A}, which often denotes the complement of A in other contexts. It’s easy to know which is meant if you think about it, but this sort of thing should be run by the cerebellum.

December 28, 2007

Length and Area I

Filed under: math.CA — Pietro @ 6:54 am
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This is a topic which has fascinated me for years, and on which I’ve spent many (fortunately unsuccessful) hours trying to prove theorems that are actually false. I find it to be a good introduction to the weirdness of analysis: you know, the gradual realization that much of our intuition about lines, surfaces and continuity is simply not applicable to the analytical formalization of lines, surfaces and continuity. Some mathematicians, like Weyl and Brouwer, blamed this weirdness on the fact that the intuitive notion of a continuum (the line) is not very well captured by something as disconnected as a set of points (the real numbers).

In any case, we have to develop some new intuitions if we’re going to expect, rather than be surprised by, the Koch snowflake, nonmeasurable sets, and families of curves that have positive area but only at the endpoints. (more…)

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