I often find in text books that the ideal gas law given by $p\cdot V=m\cdot R\cdot T$ is applicable in a closed system, still, some use it for open systems (like freezing the system at a given instance, and consider it closed). Can any one tell if this is also true? and if not? what are the errors that one might have in his calculations?
You can consider this to be true for any parcel of air, even in the open.
In the real world however, there are usually added dynamics due to convection mechanisms (i.e. winds and the sort) that screw this up.
But a good for instance would be a bubble out of a divers snorkel, in depth. the boundary for the bubble is in no way rigid, and the bubble expands as it floats up.
In the relatively still water, "winds" play a small role in deforming the bubble (in general).
In fact using the ideal gas law on an air parcel in the atmosphere is how one gets the so-called barometric equation, which is by-and-large a good approximation to how gasses in the atmosphere behave.
It is true however that a better approximation exists, but nonetheless the better approximated equation STILL uses law of ideal gasses.