Suppose $X$ is a topological space that is locally conneted and let $O$ be an open subset of $X$. Then we want to show that $O$ is also locally connected.
Let $p\in O$ chosen arbitrarily, then there exists a open set $U\subset O$ and a connected open set $V\subset X$, both of which containing $p$. Now their intersection $W=U\cap V$ is open and contains $p$, therefore $W$ is a neighborhood of $p$ and $W\subset U\subset O$. If $W$ is connected then we are done. Now, suppose that $W$ is not connected hence can be written as a disjoint union of two clopen sets $A$ and $B$. Then $A\cup B=W = U\cap V\subset V$. But since $V$ is connected, then the only clopen subsets of $V$ are $\phi$ and $V$ itself. We get the following cases:
- Case 1: $A=B=\phi$, we have a contradiction since $p\in W=A\cup B$.
- Case 2: $A=V$ or $B=V$, then $V= A\cup B=W=U\cap V\subset U\subset O$ and and since $p\in V\subset U \subset O$, therefore $O$ is locally connected since the choice of $p$ was arbitrary.
Is the proof okay? Did I make a mistake somewhere?