Standard Notations

\(\mathbb{Z}, \, \mathbb{Q}, \, \mathbb{R}, \, \mathbb{C}, \, \mathbb{H}, \, \mathbb{O}\) : the integers, rationals, reals, complexes, quaternions, and octonions.

\(\mathbb{Z}_n\) : the integers mod \(n\)

\(\mathbb{R}^n\) : \(n\)-dimensional Euclidean space.

\(\mathbb{C}^n\) : complex \(n\)-space. In particular, \(\mathbb{R}^0=\{0\}=\mathbb{C}^0\), zero-dimensional vector spaces.

\(I=[0,1]\) : the unit interval

\(S^n\) : the unit sphere in \(\mathbb{R}^{n+1}\), all points of distance \(1\) from the origin.

\(D^n\) : the unit disk or ball in \(\mathbb{R}^{n}\), all points of distance \(\leq 1\) from the origin.

\(\partial D^n = S^{n-1}\) : the boundary of the \(n\)-disk.

\(e^n\) : an \(n\)-cell, homeomorphic to the open \(n\)-disk \(D^n - \partial D^n\). In particular, \(D^0\) and \(e^0\) consist of a single point since \(\mathbb{R}^0 = \{0\}\). But \(S^0\) consists of two points since it is \(\delta D^1\).

\(\mathbb{1}\) : the identity function from set to itself.

\(\sqcup\) : disjoint union of sets or spaces.

\(\times , \, \sqcap\) : product of sets, groups, or spaces.

\(\approx\) : isomorphism.

\(A \subset B\) or \(B \supset A\) : set-theoretic containment, not necessarily proper.

\(A \hookrightarrow B\) : the inclusion map \(A \rightarrow B\) when \(A \subset B\)

\(A - B\) : set-theoretic difference, all points in \(A\) that are not in \(B\).

iff : if and only if.

There are also a few notations used in this book that are not completely standard. The union of a set \(X\) with a family of sets \(Y_i\), with \(i\) ranging over some index set, is usually written simply as \(X \cup_i Y_i\) rather than something more elaborate such as \(X \cup(\bigcup_i Y_i)\). Intersections and other similar operations are treated in the same way.

Definitions of mathematical terms are given within paragraphs of text, rather than displayed separately like theorems. These definitions are indicated by the use of boldface type for the more important terms, with italics being used for less important or less formal definitions as well as for simple emphasis as in standard written prose. Terms defined using boldface appear in the Index, with the page number where the definition occurs listed first.