Every point in topology space belongs to exactly one connected component.

Every point in topology space belongs to exactly one connected component.

Proof.

Every {${x}$} subset in $X$ is connected.Let's look at $U(x)=\cup_i{U_i}$ where $U_i$-s contain fixed $x$ point and are connected.$U(x)$ is connected.And then author says that $U(x)$ is not contained in other connected subset different than itself.Can you please explain where we got that it isn't contained in other connected subset?

If $U(x)$ was a strict subset of a connected set $C$, we know that $x\in C$, so $C$ would be one of the $C_i$'s, and then $C\subset U(x)$. That's impossible, since $U(x)\varsubsetneq C$.