Assume that $f:(X,d_{1})\to (Y,d_{2})$ is a continuous surjective map between compact metric spaces. We define another metric $d_{f}$ on $Y$ With $$ d_{f}(y_{1},y_{2})=Hd(f^{-1}(y_{1}), f^{-1}(y_{2}))$$ where $Hd$ is the Hausdorff distance. This metric is used in this post,too

Is $(Y,d_{f})$ a locally compact space? can this topological space be topologically embedded in $(Y, d_{2})$?

If the answer to the first part of the above question is affirmative, then we have the following situation in the context of commutative $C^{*}$ algebras:

We have an embedding of a (commutative) separable $C^{*}$ algebra $A=C(Y)$ into another separable (commutative) $C^{*}$ algebra $B=C(X)$ and finally we obtain a new (not necessarilly unital $C^{*}$ algebra. What is a possible non commutativization of this processes?

- Is there an example of this situation such that $f:(X,d_{1})\to (Y, d_{f})$ would be discontinuous at all points.

**Edit:** According to the counterexample of Eric, I would like to add some extra conditions to the first question as follows:

What about if each of the following conditions are assumed :

i)$f:X\to Y$ is a covering map (a covering space structure)

Or

ii) $X$ and $Y$ are smooth compact manifold and $f$ is an smooth map