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:
(
is open in
);
(
(the interior of A);
(the closure of A).
They’ve been saving me a lot of time and thought since then, like notation’s supposed to. Witness 
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 |
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.
)
Comment by Pietro — April 20, 2008 @ 11:05 pm |