Notation

Here we record the notation that we will use throughout the wiki.

Special Sets

 * $$\mathbb{N}$$ is the set of natural numbers $$0,1,2,\dots$$
 * $$\mathbb{Z}$$ is the set of integers $$\dots,-2,-1,0,1,2,\dots$$
 * $$\mathbb{Q}$$ is the set of rational numbers
 * $$\mathbb{R}$$ is the set of real number

Set Theory
We will typically use capital letters for sets and lower-case letters for elements of sets, barring some compelling reason not to.
 * We say $$A \subseteq B \Leftrightarrow \forall a \in A(a \in B)$$. We say $$A \subset B \Leftrightarrow A \subseteq B$$ and $$A \not= B$$.
 * $$\{x \in A : P(x)\}$$ is the set of elements $$x$$ in the set $$A$$ satisfying $$P(x)$$. For example $$[0,1] = \{x \in \mathbb{R} : 0 \leq x \leq 1\}$$.
 * $$X \times Y = \{(x,y) : x \in X, y \in Y\}$$ is the set of all pairs with first coordinate in $$X$$ and second coordinate in $$Y$$.
 * Similarly $$X_1 \times X_2 \times \dots \times X_n$$ is the set of n-tuples with i-th coordinate in $$X_i$$.
 * $$X^n = X \times \dots \times X$$ n times. It is the set of n-tuples with each coordinate in $$X$$
 * $$A \cap B = \{c : c \in A\ \text{and}\ c \in B\}$$ is the intersection of two sets.
 * $$\bigcap_{i \in I} A_i = \{c : c \in A_i\ \text{for each}\ i \in I\}$$ is the intersection of a family of sets.
 * $$A \cup B = \{c : c \in A\ \text{or}\ c \in B\}$$ is the union of two sets.
 * $$\bigcup_{i \in I} A_i = \{c : c \in A_i\ \text{for some}\ i\}$$ is the union of a family of sets.
 * $$B \setminus A = \{c : c \in B\ \text{but}\ c \not\in A\}$$ is the difference of $$B$$ and $$A$$ or the compliment of $$A$$ in $$B$$.
 * If $$A \subset X$$ and $$X$$ is clear from context, $$A^c = X \setminus A$$ is the compliment of $$A$$ (in $$X$$).
 * If $$f:X \rightarrow Y$$ is a function from $$X$$ to $$Y$$ and $$A \subset Y$$, then $$f^{-1}(A) = \{x \in X : f(x) \in U\}$$ is the preimage of $$U$$ under $$f$$.

Metric Spaces

 * If $$(X,d)$$ is a metric space, $$B(x,\epsilon) = \{y \in X : d(x,y) < \epsilon \}$$ is the ball of radius $$\epsilon$$ centered at $$x$$.