Let $(X, \tau)$ be a Hausdorff and compact space, given $x \in X$ we consider the collection $\mathcal{C}= \{C_i\}$ of all the closed and open subsets $C_i$ such that $x \in C_i$. Prove that $C = \cap C_i$ is connected. Prove that $C$ is the connected component of $x$ in $X$.
I tried this, but I am not sure it is right as I did not use X being Hausdorff and compact. Given $\{U,V\}$ such that $C= U \sqcup V$, then $U = C \cap X-V$, w.l.o.g $x \in U$, so $U$ is an open and closed (intersection of closed subsets) subset. Therefore, $C \subset U$ , so $V = \emptyset$ #
