I know a typical setup for a closed circuit consist of a battery, resistor, a lightbulb and a switch but that's not important, the only thing matters is we closed the circuit so that there is a difference in electrical potential somewhere but how come this voltage difference only appears after the circuit is closed? All we need to do is to move the valance electron to induce an electric field that tells another valance electron to move and this pattern should continue along the piece of conducting wire so I have no idea why must we form a closed loop for it to work?
Current flowing in a closed circuit is just a special sub-case required for continuous operation of electronic circuits. In general, current and charges do not need a closed circuit to flow. Think about things like static charges, lightning bolts, and antennas. It's just that if the circuit is not closed then charge accumulates and eventually cancels out the applied electric field at which point charges stop flowing. This happens very fast and is not of much interest in most technology.
It is not necessary for the charges to flow only when the wires are making physical contact with each other, the leakage current and the arcing are the example of the same but in the case of arcing this requires the huge potential difference between the two points, if this potential difference becomes too large then this cause the air to become ionized which makes that amount of area full of ionized particle and ionization of the air makes the air much more conductive and hence it completes the circuit and the arcing happens.
That means the charges does not require the circuit to be physically connected, and in the case of low voltage there is still a small amount of current flowing through the circuit even if it's open although it's not significant and which we call it as a leakage current.