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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