Is the property of local connectedness or local path-connectedness invariant under homotopy equivalence?

Question

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

Answer 1

No. Consider the subset of $\Bbb R^2$ $$X=\Big([0,\infty)\times\{0\}\Big)\cup\Big(\{0\}\times[0,1]\Big)\cup\left(\bigcup_{n\in\Bbb N}\{1/n\}\times[0,1]\right)$$ Which is path-connected but not locally path-connected. It deformation-retracts to $[0,\infty)\times\{0\}\cong [0,\infty)$, though.

Similar thing happens with the suspension of $\Bbb Q$: $$X=\{(tq,1-t)\,:\,q\in\Bbb Q\wedge t\in[0,1]\}\subseteq\Bbb R^2$$ It is path-connected, but not locally connected, and it deformation-retracts to the point $\{(0,1)\}$.