TITLE: An open cover characterization of connected spaces?
QUESTION [5 upvotes]: Is it true that a topological space $X$ is connected if and only if for every open cover $\{U_s\}_{s \in S}$ of $X$, and for every pair of points $x_1,x_2 \in X$, there is a finite sequence $s_1,...,s_k$ of elements of $S$ such that $x_1 \in U_{s_1} , x_2 \in U_{s_k}$ and $U_{s_i} \cap U_{s_j} \ne \emptyset$ iff $|i-j|\le1$ ?

REPLY [2 votes]: HINT: Let $X$ be a topological space. For $x_1,x_2\in X$ write $x_1\sim x_2$ if and only if there is a finite sequence as in the statement of the question.

Show that $\sim$ is an equivalence relation on $X$.  
Show that each equivalence class of $\sim$ is a union of open sets and is therefore open.  
Explain how the desired result follows from this.