Let $X$ be a topological space and suppose that $U$ is a nonempty subset of $X$ that is open, closed and connected. Does it follow that $U$ is a connected component of $X$? If not, what condition on $X$ would ensure that it is?
Attempt: Since every connected subset of $X$ intersect exactly one connected component, we have that $U$ is contained in a connected component $C$. Thus, it would suffice to have that $C$ is open and closed, so that $U$ is open and closed in $C$ and hence $U=C$.
Definition of connected component: Define an equivalence relation on $X$ by setting $x\sim y$ if there is a connected subspace of $X$ containing both $x$ and $y$. The equivalence classes are called the connected components of $X$. (Taken from Munkres.)
Best Answer
Let $V$ be a connected component such that $U\subset V$, $V-U$ is closed since $U$ is open and closed, and $V=U\bigcup (V-U)$. So either $U$ or $V-U$ is empty since $V$ is connected. We deduce that $V-U$ is empty.