Index set

From Free net encyclopedia

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 <math>J \sub \mathbb{N}</math>, where f : J -> S is the particular enumeration of S.

• Any countably infinite set can be indexed by <math>\mathbb{N}</math>.

• For <math>r \in \mathbb{R}</math> let <math>f_r : \mathbb{R} -> \mathbb{R}</math> be given by

<math>f_r(x) := \begin{cases} 0, & \mbox{if } x \ne r \\ 1, & \mbox{if } x = r. \end{cases} </math>

The set <math>\{f_r\}</math> (which happens to be a basis for the vector space of all functions on <math>\mathbb{R}</math> over <math>\mathbb{R}</math>) is an uncountable set indexed by <math>\mathbb{R}</math>.

See also

• Index
• Indexed familyde:Familie (Mathematik)

it:Famiglia (matematica) hu:Halmazrendszer no:Familie (matematikk)