Why do many Collection classes in Java extend the abstract class and implement the interface as well?

Why do many Collection classes in Java extend the Abstract class and also implement the interface (which is also implemented by the given abstract class)?

For example, class HashSet extends AbstractSet and also implements Set, but AbstractSet already implements Set.

It's a way to remember that this class really implements that interface.
It won't have any bad effect and it can help to understand the code without going through the complete hierarchy of the given class.

From the perspective of the type system the classes wouldn't be any different if they didn't implement the interface again, since the abstract base classes already implement them.

That much is true.

The reason they do implement it anyways is (probably) mostly documentation: a HashSet is-a Set. And that is made explicit by adding implements Set to the end, although it's not strictly necessary.

Note that the difference is actually observable using reflection, but I'd be hard-pressed to produce some code that would break if HashSet didn't implement Set directly.