Hi, I believe I have the right information for the following two functions but I just need a brief statement as to why it is true.

Consider the set of rational numbers.

1) , which I said is

2) , which I said is

3) , which I said is

And for the second function, S={(x,y): y = sin(1/x), x>0} I said the following:

1) , which I said is

2) , which I said is

3) , which I said is

Again, I just need brief statements verifying the validity of these statements using point/ball methods used in elementary topology. Thanks a lot, I appreciate it.