Connected preimage of quotient map

Question

I'm reading a paper on the fundamental groups of quotient spaces, and thought of the following question:

Let $f: X \to Y$ be a quotient map with $X$ locally path connected and path connected, and further assume that each fiber $f^{-1}(y)$ is connected. Is $f^{-1}(U)$ connected for each connected subset $U\subset Y$?

Any ideas would be appreciated!

Answer 1

Here's a counterexample.

Let $X=\{a,b,c,d\}$ with the topology having the base $\{a,b\},\{b\},\{b,c,d\},\{d\}$, and let $Y=\{A,C,D\}$ with the topology generated by $\{C,D\},\{D\}$. Then $$ q:X\to Y, \quad a\mapsto A,\; b,c\mapsto C,\; d\mapsto D $$ is a quotient map with connected fibers. However, $U=\{A,D\}$ is a connected set in $Y$ with disconnected preimage $\{a,d\}$