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 $\displaystyle \mathfrak{d}$(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 $\displaystyle \mathfrak{d}$(A)
Let int (A) be an interior of a set A.
Since $\displaystyle X = int (A) \cup \ \mathfrak{d}(A) \cup int(X \setminus A)$ and C intersects both A and X \ A,
$\displaystyle C=(C \cap int (A)) \cup (C \cap \mathfrak{d}(A)) \cup (C \cap int(X \setminus A))$,
Now suppose to the contrary that $\displaystyle C \cap \mathfrak{d}(A)$ is empty. If $\displaystyle C \cap \mathfrak{d}(A)$ is empty, then both $\displaystyle C \cap int (A)$ and $\displaystyle C \cap int(X \setminus A)$ are not empty (C should intersect both A and X\A). We know that int (A) and int $\displaystyle (X \setminus A)$ are not connected. It follows that $\displaystyle C=(C \cap int (A)) \cup (C \cap int(X \setminus A))$ is not connected, contradicting that C is a connected subset of a topological space X.
Therefore, C intersects $\displaystyle \mathfrak{d}$(A).