Connectedness of finite intersections

In the usual circle topology (open arcs), it seems that the intersection of a finite number of connected sets is either empty, a connected set, or the disjoint union of two connected sets.

Can we construct topological spaces in which the intersection can be the disjoint union of more than two connected sets? If so, how are these topological spaces called due to this property?

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Is circle topology topology of open balls (metric topology)? –  user123454321 Feb 2 '13 at 5:50

In $\mathbb R^2$ you can find plenty of subspaces $X,Y$ which are connected and such that the intersection $X\cap Y$ is the disjoint of any number (even infinite) of maximally connected subsets. For instance, let $Y$ be the graph of the $\sin$ function and let $X$ be the $X$-axis.
To add to that, even if one wanted to consider compact spaces, simply take $[0,1]\times\{0\}$ and intersect with the topologist's sine curve. I wasn't sure if the OP would be interested in additional restrictions, so I thought I'd put it here. –  Clayton Feb 2 '13 at 5:58
The intersection of | and S seen in the symbol \$ is the set ⋮.