The following is a problem I questioned myself yesterday:
Let $X$ a topological space and define the equivalence relation $\sim$ given by:
$$x\sim y \Leftrightarrow \textrm{ $x$ and $y$ can be connected with a continuous path.}$$
If $X/\sim$ is path connected, is it true that $X$ is path connected?
I thought about this a lot yesterday and still couldn't find any good idea to prove or disprove this.
I think the answer is no. Take for instance the subspace $X$ of $\mathbb{R}^2$ defined by $$ X = \{(x,\sin 1/x): x > 0\} \cup \{(0,y) : -1 \leq y \leq 1\}. $$ This space has exactly two path components $$ A = \{(x,\sin 1/x): x > 0\} \text{ and } B = \{(0,y) : -1 \leq y \leq 1\}. $$ So $X/\sim = \{A,B\}$. Let $\pi: X \to X/\sim$ be the quotient map. Suppose $U$ is an open subset of $X/\sim$ containing $B$. Then $\pi^{-1}(U)$ is an open set of $X$ and, therefore, $\pi^{-1}(U) \cap A \neq \emptyset$. Thus $A \in U$. On the other hand the set $\{A\}$ is open in $X/\sim$, since $\pi^{-1}(\{A\}) = A$. Thus $X/\sim$ is the Sierpiński space, i.e., its topology is $\{X/\sim, \emptyset, \{A\}\}$. This space is path connected.