What to see in or near to Copenhagen?

I am going to Denmark next June for 7-9 days and will spend most of the time in Copenhagen and Funen. What are the most interesting things to see in or close to that area?

You could spend a couple of hours in Sweden, Malmoe is close and easy to reach, just 12 minutes by train from Copenhagen. The old town is really nice and worth a visit.

You need to decide to yourself. What I want to see, what is my interest

Tivoly gardens in copenhagen also Hans Christens home along the canal which you can take a boat ride to and also the little mermaid.
There is also a church in roskille where all the king & queens are layed to rest very interesting take pictures.