Menger sponge

Example

Let $M_0 = [0, 1]^3$. Divide each cube of side $3^{-k}$ in $M_k$ into $27$ equal cubes of side $3^{-k-1}$. Remove the interiors of the central cube and of the $6$ cubes that share a face with it. The union of the other $20$ cubes is $M_{k+1}$. The Menger sponge is $M = \bigcap_{k \ge 0} M_k$.

The set $M$ is a continuum. Each $M_k$ is a finite union of closed cubes, so it is compact. It is connected: the $20$ kept cubes in each divided cube are the $12$ edge cubes and $8$ corner cubes, which form a connected frame. A decreasing intersection of nonempty compact connected sets is nonempty, compact and connected.