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.
By “being the standard topology”, I assume you the relative topology on .
I added letters to your set for reference.
Set is neither open nor closed in , however it is open in . Can you explain this?
Set is neither open nor closed in . It is not open in because it contains a boundary point and it is not closed because is a limit point not in the set.
Set is closed in , and it is closed in .
Can you explain this?