$\prod_{a\in A}D_a=\{f:A\to \prod_{a\in A}D_a:f(a)\in D_a\}$ According to my book the Cartesian product of an indexed family $\{D_a\}_{a\in A}$ is the same as the set of functions from the indexing set A such that $(\forall a\in A) (f(a)\in D_a)$.
$$\prod_{a\in A}D_a=\{f: Func(f) \land Dom(f)=A\land f(a)\in D_a\}$$
How can I get some insight into that?
p.s. The book is Kelley's General Topology. Chapter 2. After theorem 3, and before theorem 4. This statement came with no further explanation. It is assumed trivial, but I don't see it.
 A: I believe that Kelley is defining the Cartesian product $\prod_{a \in A} D_a$ to be the family you name.
But to gain some insight as to how this works, I am sure you are familiar with "finite Cartesian products" like $A \times B$, which is the set of all ordered pairs $( a , b )$ where $a \in A$ and $b \in B$.  You can additionally think of an ordered pair as a function $p$ with domain $\{ 0 , 1 \}$ such that $p (0)$ is the first coordinate of the pair, and $p(1)$ is the second coordinate.  So, in a certain sense we can consider $A \times B$ to be the family of all functions $p : \{ 0 , 1 \} \to A \cup B$ such that $p ( 0 ) \in A$ and $p ( 1 ) \in B$.
The extrapolation of this idea to the infinite is the genesis of the modern notion of a Cartesian product.
A: Try a simple example: the set $X = \{5,6,7\}$. The Cartesian product with index set $I = \{1,2,3\}$ is $\prod_{n=1}^3 X = \{ (x,y,z) : x,y,z \in X \}$.
A function from $I$ to $X$ maps every $i$ to an $x$ in $X$. It is the same as a $3$-tuple. Hence the set of all functions from $I \to X$ is $\{ (x,y,z) : x,y,z \in X \}$.
Similarly for infinite sets.
