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the term doesn't actually have direct relation to cities
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.
Battlespace determination is mentioned in IPTSF Text 27 as a form of Natural Predictamancy, because if you are in unaffiliated space, and your unit normally gets its turn at dawn, but you don't actually get your turn, then some other Side's turn is starting at dawn. Since the only side you know about is your own, then you must be in the battlespace of the other side which did get its turn. So even if you can't see them, you know they are there, and can now predict that battle is possible.