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 );
- ( is closed 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 . The closure symbol, in particular, has ended the ambiguity with , which often denotes the complement of 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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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. :o)

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