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

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

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

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

• 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

  \[
  \vect{x}: J \rightarrow X.
  \]

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.