TITLE: Topological spaces determined by generalized metric spaces
QUESTION [12 upvotes]: At http://en.wikibooks.org/wiki/Real_Analysis/Metric_Spaces you can find the standard definition of a metric space: a set $X$ given with a function $d:X\times X\to\mathbb{R}$ that satisfies properties 1 through 4. Later on the page it defines open ball and open set, and proves that arbitrary unions and finite intersections of open sets are open. (The page has a few mistakes; in particular, in the second paragraph of the proof, the unions should be intersections.) In other words, the open sets form a topology.
What I find remarkable is that none of properties 1 through 4 are needed for this proof. So consider a set $X$ and an arbitrary function $d:X\times X\to\mathbb{R}$. We can define open balls and open sets using the same definitions, producing a topology on $X$. Such topologies can be very different from those that arise from true metrics; for example, if $d$ is identically 0 we get the indiscrete topology.
Can we prove anything interesting about which topologies can arise this way? In particular, does every finite topology arise this way?

REPLY [12 votes]: Some partial answers to OP´s original question:
1) Any first-countable space (and in particular any finite topology) can be obtained in this way.
Let $X$ be first-countable, and for each $x \in X$ fix a local neighborhood base $\{B_n^x:n \in \omega \}$ such that $X=B_0^x$ and $B_{n+1}^x \subseteq B_n^x$. We can now define the (not necessarily symmetric) function $$d(x,y)=\inf \left \{\frac{1}{n+1}:y \in B_n^x \right \}.$$
It is not hard to see that the topology defined by this function (defining balls centered at $x$ as $B(x,\varepsilon):=\{y : d(x,y) < \varepsilon \}$) is the same as the original topology on $X$.
2) One can get non-first-countable topologies.
If we let $X=\{a_{n,m} : n,m \in \omega \} \cup \{b_n :n \in \omega\} \cup \{c\}$ and define $d(b_n,a_{n,m})=d(c,b_n)=\frac{1}{n+1}$, $d(x,x)=0$ and $d(x,y)=1$ otherwise, we obtain Arens´ space which is first-countable at every point except at $c$.
3) Not every topology can be obtained in this way.
If $X=D \cup \{\infty\}$ is the one-point compactification of an uncountable discrete space, then the topology of $X$ cannot come from a "generalized metric". Just note that in this situation the balls $B(\infty,1/n)$ have to be open and hence cofinite; thus their intersection is cocountable and therefore the topology induced by the generilized metric wouldn´t be Hausdorff (but the original topology is).
Edit: here is another example for 3), prompted by Marcos´ comment, which is interesting because it shows that the class of spaces that we are looking at is not closed under subspaces: 
The Arens-Fort space, which is just the subspace $\{a_{n,m} : n,m \in \omega \} \cup \{c\}$ of the Arens´ space defined in 2), can´t be obtained by this procedure. The reason is that the balls $B(x,\varepsilon)$ would have to be open (this is not hard to see) but then $\{B(c,1/n):n \in \omega\}$ whould be a countable local base at $c$.