Recover premetric from subset of topology

Question

From Is a symmetric premetric space a topological space? I am aware that a premetric $d: X \times X \rightarrow \mathbb R$ defined for elements of a set $X$ induces a topology $\tau$ on the set, so that $(X, \tau)$ is a topological space. A topology is a set of subsets (called neighborhoods $N$) of $X$, so here

$$ \tau = \{ N \mid N \subset X \} $$

Let's say I do not know $d$, but I am given some elements of $X$ and a subset of the topology, namely

$$ O = \{ N \mid N \subset X \} = \{ \{ x_1, x_2, ..., x_i \}, \{ x_{i+1}, x_{i+2}, ... \}, ... \}\subset \tau $$

where all the $x\in X$.

• Which methods / algorithms etc. exist which (approximately) recover $d$? (At least all $d(x_i,x_j)$ for all $x_i, x_j$ that I have seen in $O$?)
• If there are no methods / algorithms yet, are there topics that I would have to / should look into to be able to develop such a method? (I have been considering Hypergraphs.)
• Are there theoretical reasons why it might not be possible to derive $d$? (In that case it might not be worth trying.)

Answer 1

Given $\tau$, it is impossible to exactly recover $d$ when $X$ has more than one point. For instance, a discrete topology of a set $X$ can be generated by any premetric $d$ of $X$ such that for any $x\in X$ there exists $\varepsilon>0$ such that $d(x,y)>\varepsilon$ for each $y\in X\setminus\{x\}$.

Thus a generic problem is given a (pre)metrizable topological space $X$, construct a (pre)metric, generating topology on $X$. Usually $X$ is an infinite metrizable space, and a construction of a metric on it looks as in a following quotation from “General topology” by Ryszard Engelking (Heldermann Verlag, Berlin, 1989)).