Problem in proving d(x,y)=0 implies x=y.

[AlbertEinstein]

Hello everyone.

I am trying to prove that the in a planar domain $$U \subseteq C$$ equipped with a metric $$\rho$$, the definition of the distance between P and Q, both lying in U is given by

$$\\ \\ d_{\rho}(P,Q)=inf \left\{ L_{\rho}(\gamma): \gamma \in C_{U}(P,Q)\right\},$$​

where $$C_{U}(P,Q)$$ denotes all piecewise $$C^{1}$$-curve joining P and Q. Also $$L_{\rho}(\gamma)$$, which is the length of the curve is defined as :

$$\\ \\ L_{\rho}(\gamma)=\int_{a}^{b}\rho(\gamma(t)).\left| \gamma'(t)\ \right| dt.$$​

I have verified all the properties of a distance function, the only elusive it remains to prove that d(P,Q)=0 implies P=Q. If the infimum is attained by some curve then it is easy to see, but what if infimum is not attained? How to prove in that case? Help please.

Thanks.
Jitendra

 

[arkajad]

My idea is as follows: Suppose there are two different P,Q with d(P,Q)=0. Cover the domain with open convex regions. Then there should be two points different points P,Q with d(P,Q)=0 in one such region. In a convex region connect P and Q with a straight line segment.

But that's just an idea...