A tuple is defined as a function

In James Munkres “Topology”, the concept for a tuple, which can be (m)-tuple, (omega)-tuple or (J)-tuple, is defined from a function point of view as below.

Let (X) be a set.

• Let (m) be a positive integer and ({ 1, cdots, m }) be an index set. An (m)-tuple of elements in (X) is a function

  [
  vect{x}: { 1, cdots, m } ightarrow X.
  ]

• Let (mathbb{Z}_+) be the index set comprised of all positive integers. An (omega)-tuple of elements in (X) is a function

  [
  vect{x}: mathbb{Z}_+ ightarrow X.
  ]

• Let (J) be an index set, whose cardinality is not limited to be finite or infinite, countable or uncountable. A (J)-tuple of elements in (X) is a function

  [
  vect{x}: J ightarrow X.
  ]

For all these types of tuples, if (alpha) is an index belongs to the index set, the corresponding coordinate component of the tuple is (vect{x}(alpha)). It is written as (x_{alpha}), which is the form we often use.

From the above it can be seen that a tuple of elements, which are literally tangible data, are viewed as the rule of assignment for a function, which is more abstract. In addition, while we have already been given to the stereotype of a tuple, which is a container holding a list of ordered elements, the function mapping version of a tuple does not require any order relation prescribed for the tuple’s index set.

Considering these concepts in computer programming, a tuple of values or objects can be either stored in an ordered array as in procedural programming. Or the tuple can be stored within a function as in functional programming. Without loss of generality, this functional perspective can be further applied to matrix and tensor, which eliminates or mingles the boundary between data and operation.