If $W$ is path connected and $W \subseteq U$ with $U$ open, then there exists $V$ open and connected such that $W \subseteq V \subseteq U$?

Question

Given $X$ a topological space and $U,W \subset X$ such that $W \subseteq U$ where $U$ is open, $W$ is path connected.

Then, can we find $V\subset X$ such that $W \subseteq V \subseteq U$, $V$ is open and connected ?

Answer 1

Yes, if $X$ is locally connected. Then you can choose the union of all connected open subsets of $U$ that contain a point from $W$. (For this it suffices that $W$ is connected; it doesn't need to be path connected in particular).

On the other hand, if $X$ is not locally connected, then by definition there is a $w\in X$ such that there is a neighborhood $U$ of $w$ that does not contain any open connected neighboorhood of $w$. In that case, you get a counterexample by letting $W=\{w\}$.

As G. Sassatelli pointed out in a deleted answer, a concrete counterexample would be $X=U=\mathbb Q$, $W=\{0\}$.