## Is there a conceptual reason why topological spaces have quotient structures while metric spaces don’t?

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.

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Are you thinking about the category $\mathcal{C}$ of metric spaces and continuous maps, or the category $\mathcal{D}$ of metric spaces and metric-preserving maps (ie isometric embeddings)? In $\mathcal{C}$ (which is essentially a full subcategory of topological spaces) I don't think that there is anything interesting to say about your question. Category $\mathcal{D}$ is much more natural if you want to make analogies with algebraic categories. – Neil Strickland Mar 6 at 11:53
I'm not sure whether this really answers your question, but the reason why limits and colimits of topological spaces exist is that on a given set there are always the coarsest and the finest topology satisfying some condition. On the other hand it doesn't seem sensible to talk about "coarsest" and "finest" metrics. – Karol Szumiło Mar 6 at 12:47
On the other hand, a quotient norm is often seen for normed spaces. – Gerald Edgar Mar 6 at 13:22
I'm dubious of the motivation in the first paragraph. Plenty of structures require some conditions for a quotient to be an object of the same type. For example, modding out a group by an equivalence relation, even an equivalence relation arising from a subgroup, does not in general give a group; you need normality. Even in the case of topological spaces, where you can always define quotients (as per Karol's comment), you need conditions for the resulting space to be nice. So your question could just as easily ask about Hausdorff spaces, or many other things, instead of metric spaces. – Noah Stein Mar 6 at 13:44
@Noah: I agree in principle, but note that quotients do exist in Hausdorff spaces, as Hausdorff spaces are a reflective subcategory of all topological spaces, hence cocomplete. – David Carchedi Mar 6 at 16:12

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 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: 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).

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This kind of limit/quotient is important in the theory of group actions on $\mathbb{R}$-trees, for instance the paper of Levitt and Paulin, "Geometric group actions on trees", MR1428059. – Lee Mosher Mar 6 at 13:24
Eric, do you mean 1-Lipschitz (non-expansive) maps, or is it strictly distance-decreasing and/or strictly preserving (isometries)? It seems that what you're saying is easy to prove for 1-Lipshitz maps, but anyway are there any good references for this sort of questions? – Sergey Melikhov Mar 7 at 4:10