Let A be a subset of a topological space of X. Show that if C is a connected subset of X that intersects both A and X-A, then C intersects (A)
Let A be a subset of a topological space of X. Show that if C is a connected subset of X that intersects both A and X-A, then C intersects (A)
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, C intersects (A).