Let int (A) be an interior of a set A.

Since and C intersects both A and X \ A,

,

Now suppose to the contrary that is empty. If is empty, then both and are not empty (C should intersect both A and X\A). We know that int (A) and int are not connected. It follows that is not connected, contradicting that C is a connected subset of a topological space X.

Therefore,Cintersects (A).