My set theory is a bit rusty so here goes.

Ok, so we want a set that is closed but infinite, that is not perfect.

Closed says that all accumulation points are contained in the set.Not perfectmeans that there must be points in the set that are not accumulation points.

How about

Define an open set to be any subset of containing 0 or is the empty set. You may need to prove that this collection of sets satisfies what it means to be open.

You can show that the set of accumulation points is which is a non-empty proper (strict) subset of .