This is the first time I have come across the following definition for the infinite Cartesian product. I somewhat understand it, however, below I have pointed out where I am getting confused.
Definition for the infinite Cartesian product:
$$\prod_{i \in \mathbb{N}}\mathbb{X}_i = \{f: \operatorname{dom}(f) = \mathbb{N} \wedge
\forall i, f(i) \in \mathbb{X}_i\}.$$
Here is where I am getting confused. Define the following:
$$\mathbb{R}^2 = \mathbb{R} \times \mathbb{R} = \{(x,y): x \in \mathbb{R} \wedge y \in \mathbb{R}\} \quad (i)$$
$$\mathbb{R}^2 = \mathbb{R} \times \mathbb{R} = \prod_{i=1}^2\mathbb{R}_i = \{f: \operatorname{dom}(f) = \{1,2\} \wedge f(1) \in \mathbb{R}, f(2) \in \mathbb{R}\} \quad (ii)$$
Here is where I am getting confused. Lets say you want to write "express" the point where $x = 5$ and $y = \pi$, then using $(i)$ you would simply write $(5,\pi)$. How would you express the same for $(ii)$? In other words, I am just not sure how you would represent the same point using $(ii)$. I know that a function is, itself, a set.
Best Answer
The only thing you are confused in here is notations. $f$ in this definition is a coordinate function: it maps coordinate index to the corresponding value, and so each function $f$ represents a single point in a product set.
For example, point $(x_1, x_2) \in \mathbb{R}^2$ is exactly the function $f : \{1,2\} \to \mathbb{R}$ such that $f(1) = x_1$ and $f(2) = x_2$ (point $(5,\pi)$ may be represented as a such function $f$ with $f(1) = 5$ and $f(2) = \pi$).
One of the reasons for using such notation is that it could be easily generalised to any number of coordinates (including infinity) – all you need is just expand the domain of $f$. Take a look to the definition of infinite Cartesian product written using coordinates (like in $(i)$): $$ \prod_{i \in \mathbb{N}}\mathbb{X}_i = \{(x_1, x_2, \dots) \wedge \forall i, x_i \in \mathbb{X}_i\}. $$ It contains expression $(x_1, x_2, \dots)$ which is, in a sense, informal: it uses notation for points with finite count of coordinates (where coordinates are represented directly), for points with infinite number of coordinates (which are, actually, the infinite sequences). Using notation with coordinate functions gives you much more formal and explicit way to describe such objects; take a look at the formal definition of sequence, which uses the same construction.