I have been a member of union since October 2017. Btw. I am unemployed of course.
Thank you for any kind of response!
If one part is a prerequisite for being able to take the second part, then it is regarded as one "overall education". In this case it is only possible to receive dagpenge, if your break is over 4 weeks, AND the break is not due to summer holiday on the education institution. Seems to be like your case...the break is only due to summer-holiday. ...so if that's true...then you can not get dagpenge.
If the first part (AP degree) is not a prerequisite for being able to take the second part (Top Up), then the case will be judged after the normal rules for being eligible for dagpenge. If that's the case, then you can only get dagpenge if you are available for the labour-market, and also have registered your graduate status in the A-kasse at latest 14 days after graduation.
If you are covered by the second option, and therefor can have dagpenge, then it is correct that its possible to have Unemployment benefits ("dagpenge") while searching for job in another EU/EEA-country. This can be for a maximum period of 3 months.
However there are some requirements you must meet before you can do that, see them here:
As you can see you must have been registered in Denmark as unemployed for 1 week (if you go to look for job in your home-country, otherwise 4 weeks) BEFORE you go to the other country.
And you must be entitled to dagpenge, before you leave this country. You can only be accepted for dagpenge, if you stay in Denmark at the time you apply for dagpenge. But as mentioned above, I do not think that you are entitled to dagpenge in this summerbreak.
Anders Weber, www.a-kasser.dk