Battlespace is an area of hexes surrounding a city. When no non-allied Units are inside a city's battlespace, the Side's turn begins at dawn. In IPTSF Text 27 it is revealed that battlespace also includes the space occupied by fielded units of a side.
The exact size of Battlespace is unknown. It may extend around all fielded units as well as a Side's cities. There is some speculation (because of the pun) that the size of battlespace is related to the movement of bats. (which is a range of twenty hexes for scouting)
The battlespace of a city may depend on the move and reach of nearby units rather than being a fixed area. A possible definition is that the battlespace of a city is the set of hexes around all units which can reach the city with one turn's worth of movement (or long-range attacks, where applicable).
For example, if a stack of foreign units with Move 10 are stationed 30 hexes away from a city, their hex is not within that city's battlespace. However, if they are joined by Commander Zamussels on a Move 30 gwiffon, now the city's battlespace does include that hex.