I have the following question i cannot answer myself.
Consider two compacts $A$ and $B$ on the real line $\mathbb R$. Let $A'$ be a translation of $A$ and $B'$ a translation of $B$:
- $A' = A + a$,
- $B' = B + b$.
Suppose it is known that $A'\cup B'$ is contained in a translation of $A\cup B$, and $A'\cap B'$ is contained in a translation of $A\cap B$:
- $A'\cup B'\subset (A\cup B) + t_1$,
- $A'\cap B'\subset (A\cap B) + t_2$.
Is it always true in this case that $A'\cup B' = (A\cup B) + t_1$ and $A'\cap B' = (A\cap B) + t_2$?
I am in fact interested in the natural generalization of this question to closed compacts in arbitrary topological groups.
I can only prove it in the case $A\cap B = \varnothing$, which is fairly easy. I cannot prove it even for the real line, even if i assume that $A\cap B$ consists of a single point.
I have posted this question in a slightly modified form on MathOverflow.