# 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.

## 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

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

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