if the two parts (AP and Top-Up) can be regarded as an "overall education" and the AP is a condition for being able to continue on the Top-Up, there is a special rule in the in the legislation (§ 32 in rådighedsbekendtgørelsen), where it says, that a person only can get unemployment benefits in an unemployment period between two parts of a "overall education", if
- the unemployment period is more than 4 weeks
- the unemployment period must not be due to holiday on the education institution
- the person is not be paid salary or SU in the period
Furthermore it is off course a condition that the person is full available for the labor market, when receiving dagpenge.
I will say that your situation fall within §32, and therefore you can not get dagpenge in the period (july-august) between the AP and Top-Up because the period is only because of holiday at the school.
Anders Weber, www.a-kasser.dk