+\providecommand*{\pair}[2]{ \left({#1},{#2}\right) }
+
+% The Cartesian product of two things.
+\providecommand*{\cartprod}[2]{ {#1}\times{#2} }
+
+% The Cartesian product of three things.
+\providecommand*{\cartprodthree}[3]{ \cartprod{{#1}}{\cartprod{{#2}}{{#3}}} }
+
+% The direct sum of two things.
+\providecommand*{\directsum}[2]{ {#1}\oplus{#2} }
+
+% The factorial operator.
+\providecommand*{\factorial}[1]{ {#1}! }
+
+%
+% Product spaces
+%
+% All of the product spaces (for example, R^n) that follow default to
+% an exponent of ``n'', but that exponent can be changed by providing
+% it as an optional argument. If the exponent given is ``1'', then it
+% will be omitted entirely.
+%
+
+% The natural n-space, N x N x N x ... x N.
+\providecommand*{\Nn}[1][n]{
+ \mathbb{N}\if\detokenize{#1}\detokenize{1}{}\else^{#1}\fi
+}