Define a metric based on a topology Is their a systematic way, based on a topology meterizable $(X, t)$ to define or compute some metric $d$ on $X$ such that the open balls in $(X, d)$ is a metric of $(X, t)$?
 A: First of all, Daniel Rust is absolutely right that in general there is no 'canonical' metric on the given topological space. Nevertheless, there are many situations in topology when it is still important to find one, namely, in the case of smooth compact manifolds. Typically, one looks for Einstein metrics or metrics of constant curvature (sectional or scalar) or Kähler-Einstein metrics, provided you have a fixed conformal or complex structure, depending on the setting. Some of the most remarkable results in topology and geometry are proofs of existence and uniqueness of such metrics. Examples are: uniformization of Riemann surfaces, Mostow rigidity theorem, Thurston-Perelman geometrization theorems for 3-dimensional manifolds, solution of Yamabe's problem (Trudinger, Aubin, Schoen), Yau's proof of Calabi's conjecture, and, most recently, theorems of Tian, Donaldson, Chen and Sun on Kahler-Einstein metrics on Fano manifolds.  
Edit: Here is a couple of references: C.LeBrun 
"Optimal Metrics, Curvature Functionals, and ..."   http://www.math.sunysb.edu/~claude/madrid.pdf 
M.Berger "What is the Best Riemannian Metric on a Compact Manifold?"
http://link.springer.com/chapter/10.1007%2F978-3-642-18245-7_11#page-1
A: There is no canonical metric associated to a metrisable topological space because, if $d\colon X\times X\rightarrow\mathbb{R}$ is a metric, then so is $d_a$ defined by $d_a(x,y)=a.d(x,y)$ for some $a> 0$. If $a\neq 1$ then $d_a\neq d$ for all $X$ with more than one element.
In fact, it gets worse. We can't even define a canonical metric on $X$ 'up to a choice of constant multiple' because if $d$ is a metric, then so is $d^*$ defined by $d^*(x,y)=\min\{1,d(x,y)\}$ and so is $d'$ defined by $d'(x,y)=\dfrac{d(x,y)}{d(x,y)+1}$. Which are both bounded metrics which induce the same topology as $d$, and so if $d$ is not bounded, both $d'$ and $d^*$ are equivalent metrics which are not equal to $d$ or even equal 'up to a choice of constant multiple'.
