I am a beginner in topology and I am thinking about the following statement in my book. I do not see how I could prove this ...
$(X, \tau)$ topological space. We say $X$ is connected, if for all open disjoint sets $U$ and $V$ in $X$ with $X = U \cup V$, we have $U = \emptyset$ or $V = \emptyset$.
I am trying to show now the following equivalence:
$A \subset (X,\tau)$ connected $\Leftrightarrow \forall U, V \in \tau$ with $A \subset U \cup V, A\cap (U \cap V) = \emptyset$ it follows: $A \subset U$ or $A \subset V$
$\Rightarrow$ : The sets $U' = A \cap U, V' = A \cap V$ are open in the subspace $A$. We have $U' \cap V' = (A \cap U) \cap (A \cap V) = A \cap (U \cap V) = \emptyset$, $A = A \cap (U \cup V) = (A \cap U) \cup (A \cap V) = U' \cup V'$. Since $A$ is connected, $U' = \emptyset$ or $V' = \emptyset$ which is equivalent to $V' = A$ or $U'= A$. The latter means $A \subset V$ or $A \subset U$.
$\Leftarrow$ : Let $A = U' \cup V'$ with disjoint sets $U', V'$ which are open in $A$. There exist open $U, V \subset X$ such that $U \cap A = U', V \cap A = V'$. Obviously $A \subset U \cup V$, $A \cap (U \cap V) = (A \cap U) \cap (A \cap V) = U' \cap V' = \emptyset$. Hence $A \subset U$ or $A \subset V$ which is equivalent to $A = U'$ or $A = V'$. The latter means $V' = \emptyset$ or $U' = \emptyset$.