I am trying to show something that requires me finding a connected neighborhood of a point. This seems true, but I don't know why I am doubting myself:
Let X be a topological space. If $x \in X$, then there exists an open neighborhood $U$ of $x$ that is connected.
Can I just pick any open neighborhood and then if it's not connected, then decompose it into disconnected components and pick the one that $x$ is in. Then if it's still not connected then I can repeat. But I am afraid what if this procedure doesn't terminate? Is the statement above true in general?
From my book, the definition of a neighborhood is: $U$ is a neighborhood of $x \in X$ if there exists an open set $V$ such that $x \in V \subset U \subset X$.