Equivalence relation on set of all metrics of a fixed set $M$

Question

Let $M$ be any set. For two metrics $d$ and $d'$ on $M$ consider the relation $$d \sim d' :\Leftrightarrow \mathcal{T}_d = \mathcal{T}_{d'}$$ where $\mathcal{T}_d$ denotes the topology on $M$ induced by the metric $d$.

Sometimes, it is not possible to show that something is an equivalence relation since the underlying set is too large or cannot be considered as a set (the set of all groups for example). Is it in this explicit case possible to say that the above relation is an equivalence relation on the set of all metrics on $M$?

Answer 1

Yes, the set of all metrics on a fixed set $M$ is a subset of the set of all functions $M \times M \to \mathbb R$, which is a subset of the set of all relations on $M \times M \times \mathbb R$, which is $\mathcal P(M \times M \times \mathbb R)$. This is under any ordinary set theory a set.

As a general intuition, any collection that is somehow bound to a finite number of objects (such as a set of functions from $M \times M \to \mathbb R$) forms a set, while collections which have some unbounded components (such as "all sets" or "all groups" or "the functions from $M$ to any other set") are cause for concern.