$X$ is a topological space, and for each point $x\in X$, it has a path connected neighborhood. Can $X$ be not locally path connected?
I think this is possible since for locally path connected, we require that for any neighborhood $U$ of $x\in X$ should contain a path connected neighborhood $V$, no matter how "small" the $U$ is.
However, if just let every point has a path connected neighborhood $V$, it's possible that there exists a neighborhood of $x$, let's say $U$, such that $U\subset V$, but $U$ doesn't contain any locally path connected neighborhood of $x$
I can't find an example. Any help on this? Thanks
