# Questions about pseudo metric on quotient space

Let $$(X,d)$$ be a metric space and $$\sim$$ be an equivalence relation on $$X$$, then we can form the quotient space $$X/\sim$$. We can also define a pseudo metric on the set of equivalence classes as follows: given two equivalence classes $$[x]$$ and $$[y]$$, we define $$d'([x],[y]) = \inf\{d(p_1,q_1)+d(p_2,q_2)+\dotsb+d(p_{n},q_{n})\}$$ where the infimum is taken over all finite sequences $$(p_1, p_2, \dots, p_n)$$ and $$(q_1, q_2, \dots, q_n)$$ with $$p_1\sim x, q_n\sim y, q_i\sim p_{i+1}, i=1,2,\dots, n-1$$. I guess this definition is the same as considering the complete graph whose vertex set is $$X/\sim$$, put weight $$\inf\{d(z,w)\mid z\sim x, w\sim y\}$$ on the edge connecting $$[x]$$ and $$[y]$$, and declare the distance between $$[x]$$ and $$[y]$$ to be the infimum of length of paths from $$[x]$$ to $$[y]$$.

It can be shown that $$d'$$ is a pseudo metric and the topology it induces is coarser than $$X/\sim$$, i.e., it contains fewer open sets. My questions are:

1. When is $$d'$$ compatible with $$X/\sim$$? I can see they are compatible when $$X/\sim$$ is compact and the topology induced by $$d'$$ is Hausdorff, namely $$d'$$ is a metric, in which case the identity map from $$X/\sim$$ to topology induced by $$d'$$ is a homeomorphism. The answer to this post seems to prove that $$X/\sim$$ is metrizable if $$X$$ is compact and $$X/\sim$$ is Hausdorff, but I do not see whether the metric can be taken to be $$d'$$.

2. Consider a collection of circles $$S_n$$, each of circumference $$1$$ with usual metric (the length of great arc connecting two points). The disjoint union $$\bigsqcup_n S_n$$ is metrizable by declaring points from different circles have distance $$2$$. Take the wedge sum $$\bigvee_n S_n$$. It seems to me that the pseudo metric $$d'$$ on $$\bigvee_n S_n$$ is a metric, so it cannot be compatible with the quotient topology which is not first countable. It does not seem like the Hawaiian earring either. What is it?

• nitpick: less -> fewer – Henno Brandsma Feb 24 '20 at 23:36