Consider a countable collection $I_n$ of closed connected disjoint intervals on $\mathbb{S}^1$. When this collection is maximal, the set $\bigcap \nolimits_{i=1}^{n}( \mathbb{S}^1 \backslash \bigcup \nolimits_{i=1}^{n} Int(I_n)\big )$ is the Cantor Space so such collection cannot exhaust the circle.
Is the result of collapsing each interval separately again $\mathbb{S}^1$?
