After removing a point from a connected set in sphere, are the connected components still connected when the original point is added back

Question

It can be written as theorem below (It may not true)

If $X$ is a subset of $S^2$ and suppose $X$ is connected, $x_0$ is a point of X, we consider the set $X\setminus\{x_0\}$, if $X\setminus\{x_0\}$}=$\bigcup_{i\in I} C_i$ (decomposition into connected components), whether it's true that $C_i\cup \{x_0\}$ is connected in $S_2$

I have this problem because if it's true, it can help me understand the relation of connected sets on the $\overline C$ and connected set on $C$