Exhaustions of product subsets by smaller product subsets

Question

Let $X$ be a compact metric space, $A,B\subset X$ be subsets and $f\colon X\times X\to \mathbb{R}$ a continuous function that is strictly positive on $A\times B$. Do there exist increasing sequences of subsets $A_1\subseteq A_2\subseteq \dots$ and $B_1\subseteq B_2\subseteq \dots$ such that:

• $A=\bigcup_{n\in\mathbb{N}}A_n$ and $B=\bigcup_{n\in\mathbb{N}}B_n$;
• for every $n\in\mathbb{N}$, the restriction of $f$ to $A_n\times B_n$ is bounded away from zero?

We can also rephrase the question as follows. Let $X,A,B$ as before and let $C\subset X\times X$ be a closed subset such that $(A\times B)\cap C= \emptyset$. Do there exist increasing sequences of compact sets $K^A_1\subseteq K^A_2\subseteq \dots$ and $K^B_1\subseteq K^B_2\subseteq \dots$ such that:

• $A\subseteq\bigcup_{n\in\mathbb{N}}K^A_n$ and $B\subseteq\bigcup_{n\in\mathbb{N}}K^B_n$;
• $(K^A_n\times K^B_n)\cap C = \emptyset$ for every $n\in\mathbb{N}$?

If true, this sounds like a useful lemma and I would not be surprised if it already appeared somewhere else. I need this fact (or something similar) to reduce some statement about operators on product spaces to the compact case, but I thus far failed to find a proof or a counterexample. Any suggestion?

Answer 1

Take $A,B$ complementary dense subsets of $X$ (say, rationals and irrationals in $[0,1]$). $d(a,b)$ is strictly positive on $A\times B$.

Suppose $A$ increasing join of the $A_n$, dually $B_n$, with positive distance between $A_n,B_n$ i.e. between their closures $A'_n,B'_n$.

Then Baire assures that for one $n$ (and then the successive ones), $A'_n$ (or $B'_n$) contains a open set. By density, it contains a point $b$, which must be in some $B_m$, and so for index $j$ greater than $n,m$ $A'_j,B'_j$ i.e. $A_j,B_j$ cannot have positive distance.

Corollary: in any partition as above, at least one set cannot be $\sigma$-compact.