I have to prove that for every space $X$ and it's diagonal $\Delta$ we have: $$\Delta\in P(X)\otimes P(X)\Leftrightarrow |X|\leq\mathfrak{c}.$$ For the $\Rightarrow$ I did it like this:
Suppose that $\Delta\in P(X)\otimes P(X)$. Then there must exist a countable family $\mathcal{K}\in P(X)$ such that $\Delta\in\sigma(\lbrace A\times B:\ A,B\in\mathcal{K}\rbrace)$. Consider $g:X\rightarrow P(\mathcal{K})$ that maps every $x\in X$ to such $K\in\mathcal{K}$ that $x\in K$. All we need to prove is that $g$ is injective. Suppose it's not - then there exists $y\neq x$ such that for every $K\in\mathcal{K}$ we have $x\in K\Leftrightarrow y\in K$. So we cannot separate pairs $\left< x,x\right>$ and $\left< y,y\right>$ in every $A\times B$ because $\lbrace x,y\rbrace\subseteq\mathcal{K}$ or $\lbrace x,y\rbrace\cap\mathcal{K}=\emptyset$. The contradition proves that $g$ is injective and from that $|X|\leq P(X)=\mathfrak{c}$.
Is that correct? Also I don't have any idea how to prove the second implication so I will be grateful for any hint.