Index Set

An index set is a set whose elements are used to label or index the members of another collection.

Definition

Let $\{A_i\}_{i \in I}$ be a family of sets. The set $I$ is called the index set, and each $i \in I$ is called an index. The sets $A_i$ are said to be indexed by $I$.

Examples

1. Finite index set: $\{A_1, A_2, A_3\}$ is indexed by $I = \{1, 2, 3\}$

2. Countable index set: A sequence $(a_n)_{n \in \mathbb{N}}$ is indexed by $I = \mathbb{N}$

3. Uncountable index set: The collection of all open intervals $\{(a, b) : a, b \in \mathbb{R}, a < b\}$ can be indexed by $I = \{(a,b) \in \mathbb{R}^2 : a < b\}$

Usage

Index sets appear throughout mathematics when dealing with:

• Indexed families of sets: $\{A_i\}_{i \in I}$
• Arbitrary unions: $\bigcup_{i \in I} A_i$
• Arbitrary intersections: $\bigcap_{i \in I} A_i$
• Sequences and nets
• Product spaces: $\prod_{i \in I} X_i$

Related

• countable
• cardinality