Open and closed sets Analyssi

Just write down a basis for the topology of each and it becomes obvious.

I mean, do you know what a discrete metric is? I can't imagine so if you don't know whether that set is open or closed or otherwise. Look up the definitions of the metrics you are using.

I didn't mean to post the discrete metric question as I know this is both open and closed....


but I am really unsure of how to answer parts 3 and 4.. Especially 4..-any help in pointing me in the right direction would be great!

What are you having trouble with in particular?

Do you know what it means for a set to be closed or open in each of those spaces? Can you visualise what the set of 3 and 4 look like?

I'm completely lost and don't know where to even start with either of them!

Ok, let's have a look at 3.

To check whether B is closed, you need to take an arbitrary sequence of points in B converging to a limit, and check to see if that limit must be in B - if the limit must be in B, then B is closed. To check if B is open, you need to see whether every point of B is contained in an open set which itself is entirely contained in B. Does that make sense? There are other ways of looking at openness/closedness but these are fine for now.