Show that the intersection of two connected sets is connected if the two sets are disjoint. Is the set $1\leq x^2+y^2+z^2 \leq 9$ connected and/or compact?
I think its compact because it's closed and limited in $\mathbb{R}^3$. (Hausdorff space)
2026-03-31 23:28:16.1774999696
Show that the intersection of two connected sets is connected if the two sets are disjoint.
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It is compact and connected.
Your justificative for compactness is ok. A rigorous person could only ask why the set is closed and limited. For connectedness, you can argue that the set is path-connected. Given two points in there, you can find a path connecting them (hint: spherical coordinates, and think geometrically).