TITLE: Is the intersection connected if union is simply connected?
QUESTION [9 upvotes]: It is well-known that intersection of connected sets is not necessarily connected. A standard example is two half-circles, which intersect in a discrete set $\{a,b\}$.
However, Mayer-Vietoris says that if we replace things by path-connectedness and suppose that $X=\mathrm{int}(A)\cup \mathrm{int}(B)$ is simply connected, then the result holds. 
More precisely, given sets $A, B \subset X$ such that $A,B$ are path-connected and $X=\mathrm{int}(A) \cup \mathrm{int}(B)$ is simply connected, then $A \cap B$ is also path-connected (note the problem with the standard example: the circle is not simply connected).
My question therefore is: does the above hold for connectedness? Explicitly,
Given sets $A, B \subset X$ such that $A,B$ are connected and $X=\mathrm{int}(A) \cup \mathrm{int}(B)$ is simply connected, is $A \cap B$ connected?

REPLY [7 votes]: No this is not true. You can modify the standard example of two half circles with "half circles" of the topologist's circle. This ensures that the union is simply connected but you get the same result as with the usual half circles.