2
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

$\mathscr{P}(A \times A)$ 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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.