7.1: A- Basic Notation and Terminology
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We quickly review some basic notation used in this book that is perhaps not described elsewhere. We use C, R for complex and real numbers, and i for imaginary unit (a square root of −1). We use N={1,2,3,…} for the natural numbers, N0={0,1,2,3,…} for the zero-based natural numbers, and Z for all integers. When we write Cn or Rn we implicitly mean that n≥1, unless otherwise stated.
We denote set subtraction by A∖B, meaning all elements of A that are not in B. We denote complement of a set by Xc. The ambient set should be clear. So, for example, if X⊂C naturally, then Xc=C∖X. Topological closure of a set S is denoted by ¯S, its boundary is denoted by ∂S. If S is a relatively compact subset of X (its closure in X is compact) or compact, we write S⊂⊂X.
A function with domain X and codomain Y we denote by f:X→Y. The direct image of S by if is f(S). The notation f−1 for the inverse image of sets and single points. When f is bijective (one-to-one and onto), we use f−1 for the inverse mapping. So f−1(T) for a set T⊂Y denotes the points of X that f maps to T. For a point q, f−1(q) denotes the points that map to q, but if the mapping is bijective, then it means the unique point mapping to q. To define a function without giving it a name, we use x↦F(x),
To define X to be Y rather than just show equality, we write Xdef=Y.