Let $X$ be a nonempty open subset of $\textbf{R}^n$. Define an equivalence relation $\sim$ by $x \sim y$ if and only if there is a continuous path from $x$ to $y$ contained in $X$. Show the following:
1) $\sim$ is an equivalence relation on $X$,
2) any equivalence class for $\sim$ is an open set in $\textbf{R}^n$, and
3) any equivalence class is relatively closed in $X$.
I was able to show (1) and (2). I am struggling with (3). I am trying to show that there exists a closed set $Y$ in $\textbf{R}^n$ such that $X \cap Y$ is an equivalence class, but seem to be missing something.
Hint: Let $A$ be an equivalence class. We need to show that $X\setminus A$ is an opens set in $X$. Can we write $X\setminus A$ as a union of open subsets of $X$?