Semi-Riemannian Geometry

Appendix A Notation and Set Theory

In this appendix, we recall some of the elements of set theory.

Let X be a set, and let S ₁ and S ₂ be subsets of X. We say that S ₁ and S ₂ intersect or overlap if S ₁ ∩ S ₂ ≠ ∅, and that they do not intersect, do not overlap, or are disjoint if S ₁ ∩ S ₂ = ∅, where ∅ denotes the empty set. The difference between S ₁ and S ₂ (in that order) is the set

Let {S _α  : α ∈ A} be a collection of subsets of X, where A is some indexing set. The union and intersection of the S _α are denoted by

respectively. The (Cartesian) product of sets X ₁, …, X _n is defined by

and each (x ₁, …, x _n ) is called an n‐tuple. For a set X, we denote X × ⋯ × X [n copies] by X ⁿ .

Theorem A.1

If X is a set and {S _α  : α ∈ A} ∪ {T} is a collection of subsets of X, then

1. (a)
   
2. (b)
   
3. (c)
   
4. (d)

Theorem ...