Are indexed symbols just evaluated functions?

Question

In math notations we often uses indices, of which examples are

• for indexing elements of a set as in $x_i\in X$, where $X$ is a set
• for indexing elements of a tuple as in $(x_1, x_2,...x_n)$
• as an index that is not an integer $(\bigcap_{i} U_i) \in \tau, i\in\mathbb R, U_i\subseteq X$, (when defining a topology $\tau$)
• as - some sort of parameter - that describe something, as when defining a ball $B_d(x, r) = \{ d(x,r)\ |\ r\in\mathbb R, x\in X, r>0 \}$

In the first example, the index just makes sure we can keep the $x$ apart, but there is a countable number of elements. In the second example the index indicates an order of elements. In the third example the index is not even countable anymore (since real numbers are not countable) and in the last example, the index is not even a number anymore, but a function.

This begs the first question: Is there any coherent definition of what an index represents or is just a question of gusto of the one introducing the respective notation? Can the index be used for anything?

Now, I have read in multiple contexts that an indexed 'something' can actually just be seen as an evaluated function. A very common example are time series, which basically are finite sequences, of which one can be represented by a tuple $(y_1, y_2, ..., y_n)$. Here we can say that we are considering the function $y(i)$, for which we can - just for example - calculate a regression model. I chose this example, because in fact, here we very often see it as a function of time (which is discretion and thus represented by the index $i$). But in all the above examples it seems very reasonable to also consider the respective elements as evaluated functions:

• $x_i\in X$ could be expressed as $x(i) \in X, i\in\mathbb Z$
• the elements in $(x_1, x_2,...x_n)$ could be expressed as $x(i) \in X, i\in\mathbb Z$
• $(\bigcap_{i} U_i) \in \tau, i\in\mathbb R, U_i\subseteq X$ could be expressed as $(\bigcap_{i} U(i)) \in \tau, i\in\mathbb R, U(i)\subseteq X$
• $B_d(x, r) = \{ d(x,r)\ |\ r\in\mathbb R, x\in X, r>0 \}$ could be expressed as $B(x, r | d) = \{ d(x,r)\ |\ r\in\mathbb R, x\in X, r>0 \}$ or just $B(x, r, d) = \{ d(x,r)\ |\ r\in\mathbb R, x\in X, r>0 \}$

So my second question: How are these any different? Is there a fundamental difference in expressing something as an index or as a function value? Is it just a question of preference and/or convention or is there a special meaning behind having an index in opposite to a function?

Especially in the later two examples it seems weird that an index has been used.

Answer 1

1. It’s not so much a question of “gusto” as one of legibility and convention.
2. You can define anything you want. Math notation is kinda cool that way.

Answer 2

I think that https://en.wikipedia.org/wiki/Function_(mathematics)#Index_notation explains it so well, that there is little to add:

Index notation

Index notation is often used instead of functional notation. That is, instead of writing $f(x)$, one writes $f_{x}$.

This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element $f_{n}$ is called the nth element of sequence.

The index notation is also often used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map $x\mapsto f(x,t)$ would be denoted $f_{t}$ using index notation, if we define the collection of maps $f_{t}$ by the formula $f_{t}(x)=f(x,t)$ for all $x,t\in X$.