Thread: Connected subset of a Topology Question

1. Connected subset of a Topology Question

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 $\mathfrak{d}$(A)

2. Originally Posted by flaming
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 $\mathfrak{d}$(A)
Let int (A) be an interior of a set A.

Since $X = int (A) \cup \ \mathfrak{d}(A) \cup int(X \setminus A)$ and C intersects both A and X \ A,

$C=(C \cap int (A)) \cup (C \cap \mathfrak{d}(A)) \cup (C \cap int(X \setminus A))$,

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

Therefore, C intersects $\mathfrak{d}$(A).