Index set

In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index set. The indexing consists of a surjective function from J onto A and the indexed collection is typically called an (indexed) family, often written as (A_j)_j∈J.

Examples

An enumeration of a set S gives an index set $J \sub \mathbb{N}$ , where $f:J \rarr \mathbb{N}$ is the particular enumeration of S.

Any countably infinite set can be indexed by $\mathbb{N}$ .

For $r \in \mathbb{R}$ , the indicator function on r, is the function $\mathbf{1}_r\colon \mathbb{R} \rarr \mathbb{R}$ given by

\mathbf{1}_r (x) := \begin{cases} 0, & \mbox{if }  x \ne r  \\ 1,  & \mbox{if } x = r. \end{cases}

The set of all the $\mathbf{1}_r$ functions (which happens to be a basis for the vector space of all functions on $\mathbb{R}$ over $\mathbb{R}$ ) is an uncountable set indexed by $\mathbb{R}$ .

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Examples

See also