I'm stuck on a question. It states to consider the set as a subspace of , it being the standard topology. And now I have to consider which are open in and which are open in .

The first set is , which I think is . Now, that is open in , because it is the union of two open sets in . But is it open in ? How do I tell if it is? How about the following cases:

My guess is that they are all closed in , but all open in . Am I right? Why/why not?

Thanks in advance.