Let $X$ be a topological space and let $C$ be a connected subset of $X$.
Let $\{X\_\alpha\}$ be a collection of open sets that are disjoint one another, whose union is $X$.
(That is, $X\_\alpha$ form a partition of $X$ where $X\_\alpha$ is open for each $\alpha$.)

Does it follow there exists $\alpha$ such that $C\subset X\_\alpha$?

This is the end of the question, but the below is where the question come from.

> 
> [Munkres, *Topology* 2ed, Section 54, Lemma 54.2]
> 
> 
> $C$ is a connected set, $\tilde F:I\times I\to E$ is a continuous function(it will be a lifting of $F$.).
> $\{V\_\alpha\}$ is a partition of a subset $p^{-1}(U)$ of $E$, where $V\_\alpha$ is open for all $\alpha$.
> The book says '$\tilde F(C)$ is connected and must lie entirely within one of the sets $V\_\alpha$.'
> 
> 
> 

[Added]
Here is my attempt.
Suppose not.
Then there exists $X\_0$ such that $C\cap X\_0$ and $C\setminus X\_0$ are both nonempty.
How do they form a separation of $C$?