Another way of looking at Brian's analysis above is to calculate the the maximum number of points that can be removed so that the resulting space is still connected.
Let $A = \{ x_1 , \ldots , x_n \}$ with $x_1 < \ldots < x_n$, and let $*$ denote the point of $\mathbb{R} / A$ corresponding to the identification of the points of $A$.
Note that you can remove $n-1$ points from $\mathbb{R} / A$ and leave the space connected: for $i < n$ let $z_i = \frac{x_{i}+x_{i+1}}{2}$.
On the other hand, you cannot remove $n$ points and leave the resulting space connected. If $z_1 < \ldots , z_n$ are points removed, there are four cases:
- If $z_j = *$ for some $j$, then clearly the space $( \mathbb{R} / A ) \setminus \{ z_1, \ldots , z_n \}$ is disconnected.
- If $z_1 < x_1$, then $(-\infty,z_1)$ is a clopen subset of $( \mathbb{R} / A ) \setminus \{ z_1, \ldots , z_n \}$.
- If $z_n > x_n$, then $(z_n , +\infty )$ is a clopen subset of $( \mathbb{R} / A ) \setminus \{ z_1, \ldots , z_n \}$.
- Otherwise there must be an $i < n$ is such that $x_i < z_j < z_{j+1} < x_{i+1}$ for some $j$. It is easy to see that $(z_j,z_{j+1})$ is a clopen subset of $( \mathbb{R} / A ) \setminus \{ z_1, \ldots , z_n \}$.