Why the current doesn't flow in an open branch? 
I know that the current will no flow towards the point A since there is no potential difference between A and the corresponding junction. However, between C and B, there is potential difference, so how can we say that no current flow from C towards B?
 A: When you turn that 10 V battery on, think of electrons leaving the negative pole and electrons entering the positive pole.
On it's left-hand-side positive pole, electrons are entering and the whole branch becomes overall more positive (higher potential). Soon, though, the potential is so high, that no more electrons wants to leave. No more are entering from the open end, so the "void" of missing electrons soon "holds on" to remaining electrons.
When this potential is high enough to hold back against the positive battery pole (when the potential equals the battery pole potential), then electrons stop moving. The current flow stops.
This will always happen in open circuits. You need a closed loop to have a steady current, because you can't have any points on the wires being depleted (or accumulating). Any such places will only grow/decrease in potential while being depleted/accumulated until a potential is established to balance out the battery.
A: 
there is potential difference, so how can we say that no current flow
  from C towards B?

First, the fact that there is a potential difference across two nodes of circuit does not necessarily imply that there is a current through.  On the other hand, if there is a potential difference across a resistor, there must be a non-zero current through.  So, I suspect that you're thinking about Ohm's law when you write this even though, e.g., a battery does not obey Ohm's law.
Second, a (steady) current from C towards B would violate Kirchhoff's Current Law (KCL) which, based on the conservation of electric charge, states:

At any node (junction) in an electrical circuit, the sum of currents
  flowing into that node is equal to the sum of currents flowing out of
  that node

Thus, if there were a (non-zero) current from C towards B, the must be some other current towards C such that KCL is satisfied.  But the only path towards C is from B!  In other words, by KCL, there is zero current from C towards B.
A: An easy way to solve this problem is think about the gap between A and B as a small capacitor.
In the steady state, this capacitor will be charged to the voltage difference between nodes A and B, which can be calculated using Kirchoff's rules, and the voltage on the charged capacitor will oppose further current flow.
Solving the circuit for DC and using point C as a zero reference point, we'll get $V_C=0V$, $V_B=10V$ and $V_A=10.2V$. So, the $C_{AB}$ capacitor will be charged to $0.2V$ as shown on the modified circuit below.

