If we have path-connected spaces $A_0 \supseteq A_1 \supseteq A_2 \supseteq \ldots$, is $\bigcap^\infty A_i$ path-connected?
I was thinking that if we take $A_i$ to be a $1/i$-neighborhood of the Koch snowflake $K$, then all the $A_i$ are path-connected and their intersection is $K$...but it seems clear from Is Koch snowflake a continuous curve? that $K$ is a path in the plane and is therefore path-connected. So that doesn't help.
I'm especially interested in subsets of $\mathbb{R}^n$. This question was inspired by Partitions of $\mathbb{R}^2$ into disjoint, connected, dense subsets.
I'm also curious about the same question with "connected" instead of "path-connected".