Functions on cardinals: Can we think about this as a product? In one of my Analysis homework assignments, my professor stated the following: Let $I$ be the closed interval $[0,1]$ and let $B(I)$ be the set of all (possibly discontinuous) functions from $I$ to $I$. If we consider $I$ to be an index set, and we set $X_i = I$ for every $i \in I$ we can form the product space $X = \prod_i X_i$ equipped with the product topology. In an evident way, we can define $B(I)$ with $X$, and so give $B(I)$ the corresponding topology, which we call the product topology.
While I'm still not confident about how we can identify $B(I)$ with $X$, I think I understand at least some of the broader ideas.
Meanwhile, in my Set Theory class, we just learned about cardinals. In class, if $\kappa,\lambda$ are cardinals, we write $^\lambda\kappa$ to denote the set of functions from $\lambda$ to $\kappa$.
Here's my question: If I consider $\lambda$ to be an index set, and for each $l \in \lambda$, let $\kappa_l = \kappa$, can I form a product space,
$$
K = \prod_l\kappa_l
$$
equipped with the product topology, and identify it with $^\lambda\kappa$ for any such cardinals? If so, what are the consequences of doing this? Do I learn anything useful?
Because the topic of cardinals is so new to me, I'm primarily interested in an explanation of why I either can or cannot form the above product space, so that I can then go and try and work out (at least a sketch of) a proof.
 A: Instead of $I$ being $[0,1]$ what if the let $I = \{1,2\}$? Well there are four functions $f:I\to I$ and they are
$$f_1: 1\mapsto 1, \\\quad2\mapsto 1$$
$$f_2: 1\mapsto 1, \\\quad2\mapsto 2$$
$$f_3: 1\mapsto 2, \\\quad2\mapsto 1$$
$$f_4: 1\mapsto 2, \\\quad2\mapsto 2$$
Now what is $I\times I$? Well it is the following set with four elements
$$I\times I = \{(1,1),(1,2),(2,1),(2,2)\}$$
Do you see a way here to identify $I\times I$ with $B(I)$?
More generally, if $I$ is an infinite set and $X_i$ is a collection of sets indexed by $i$, I believe you would define $\prod_{i\in I}X_i$ to consist of the elements $(x_i)_i$ such that $x_i\in X_i$?
Now let $(x_i)_i\in \prod_{i\in I} X_i$ and define a function $f:I\to \prod_{i\in I} X_i$ as follows:
$$f(i) = x_i$$
This gives us an identification of functions $f:I\to \prod_{i\in I} X_i$ such that $f(i)\in X_i$ (really the reverse, but can you see this direction as well)?
All this is to say that $I$-tuples of $A$ and functions $I\to A$ are really kind of the same thing (at least in a canonical way). This should answer your question with respect to cardinals, but I can elaborate on anything if need be.
