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.

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2 Comments »

  1. that is a good idea…

    but the “filled ball” symbol might be a little boring to make it by hand.

    for subsets, I find it quite annoying to use \subseteq for usual inclusion, I find it better to use \subset, and in case of proper inclusion, I use \subsetneq.

    using A^c for complement and \bar{A} for closure, I suggest using A \subset^\circ X for open sets and A \bar{\subset} X for closed sets.

    Comment by Gabriel Haeser — April 20, 2008 @ 1:34 pm | Reply

  2. Gabriel, thanks for your comments and suggestions!

    I suppose one of the motivations for the “filled ball” symbol is to make it more similar to the symbol for “openness” — I never liked the big typographical difference between the notations for “interior” and “closure”. It does take a little longer to write by hand, but general topology rarely fills pages and pages of calculations, so I find it to be manageable.

    Also, I’m much more a “colorful shapes” than “letters” kind of guy: my variables for big, complicated expressions tend to be black/white circles, squares and triangles, or tiny drawings of what the expression represents. :o)

    Comment by Pietro — April 20, 2008 @ 11:05 pm | Reply


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