# How the mapping from power set of a set defined?

So sorry for such an easy and bad question....

In Huber's paper page 462 , I can't understand how $$P(A\times A)=\{0,1\}^{A \times A}$$ is defined? Can some one please give an example?

$$\mathscr{P}(A \times A)$$ is just the set of all subsets of $$A \times A$$ (so in fact all relations on $$A$$; a relation on $$A$$ is precisely a subset of $$A \times A$$) and Huber wants to give this a topology: a standard identification that's often made is to associate a subset $$R$$ of $$A \times A$$ with its characteristic function $$\chi_R: A \times A \to \{0,1\}$$ defined by $$\chi_R((a,a')= \begin{cases} 1 & (a,a') \in R\\ 0 & (a,a') \notin R\end{cases}$$
It's easy to see that $$\mathscr{P}(A \times A)$$ is in bijection with $$\{0,1\}^{A \times A}$$, where the latter is the set of all functions $$A \times A \to \{0,1\}$$. He sees the latter as a space in the product topology (where we have $$A \times A$$ many copies of $$\{0,1\}$$, as it were), and each $$\{0,1\}$$ is discrete.