Of the mathematical objects that I am familiar with, it is normally the case that the product of 2 objects is an object of the same type and that an equivalence relation on an object induces a quotient object of the same type.


I think I have some understanding as to why the product of 2 fields is not a field, because a field is not an algebra in the universal algebra sense. But I don't see a reason as to why an equivalence relation on a metric space fails to induce a quotient structure, apart from the fact that it just doesn't work.



Is there a conceptual reason why topological spaces have quotient structures while metric spaces don't?
You *can* define a (pseudo)metric on a quotient of a metric space. Let $X$ be a metric space with metric $d$ and an equivalence relation $\sim$. Say that a chain between two points $x,y\in X$ is a sequence of points $x=a\_0\sim b\_0$, $a\_1\sim b\_1$, $\ldots$ $a\_n\sim b\_n=y$, and define the length of such a chain to be $\sum d(b\_i,a\_{i+1})$. We can now define the distance $d([x],[y])$ between two equivalence classes to be the infimum of all lengths of chains from $x$ to $y$.


It's easy to see that this is a pseudometric on $X/{\sim}$ (a metric where the distance between two distinct points might be $0$). This descends to a true metric on the quotient $Y=X/{\sim}'$, where $x\sim' y$ if $d([x],[y])=0$. Furthermore, $Y$ can be characterized by the following universal property: (non-strictly) distance-decreasing maps from $Y$ to a metric space $Z$ are naturally in bijection with distance-decreasing maps from $f:X\to Z$ such that $f(x)=f(y)$ whenever $x\sim y$.


More generally, a similar construction shows that the category of metric spaces and distance-decreasing maps has all connected colimits (colimits over connected diagrams). If you generalize metrics to allow the distance between two points to be infinite, you can construct all colimits, and also all limits (use the sup metric on products).


