Let $X$ be a topological space, $x\in X $, $C$ is a connected component of $x$.
Define $A$ to be the intersection of all the open-and-closed sets that contain $x$ (also called the pseudo-component sometimes).
I wish to show that $A=C$, if $X$ is also compact (without it I think there is a counter-example).
Obviously $C$ is contained in $A$ (always) since any clopen set that contains $x$ contains $C$. Also, $A$ is closed in $X$ so if $X$ is compact, $A$ is also compact. Not sure on how to proceed (tried supposing that $A$ is not contained in $C$ and getting a cover of $A$, doesn't seem to work).
Any help will be appreciated!