Is the property of local connectedness or local path-connectedness invariant under homotopy equivalence? I.e. If $X,Y$ are homotopically equivalent Hausdorff topological spaces such that $X$ is locally connected / locally path connected, then is it true that $Y$ is also locally connected/locally path connected?
I only know that connectedness, path connectedness, simply connected are all homotopy invariant; compactness, separablity (having countable dense subset) are not homotopy invariant.
Please help. Thanks in advance