Sen's property $\alpha$ holds when limited attention in choice?
My math is a bit rusty, but I believe you can show that, for a well-behaved (e.g., open and connected) feasible set $T$ and a consideration set $T'$, $x \in S \subseteq T' \subseteq T$, e.g. Sen's property $\alpha$ holds.

That doesn't seem like a very interesting question/result though, as it just follows directly from the definition of "well-behaved".  Maybe I'm missing something.