I'm new here and I'm dealing with some topology questions.
Suppose that (X,T) is a C1 space then the following statements are equivalent:
(1) (X,T) is a hausdorff space
(2) Every convergence sequence has a unique limit
(3) (X,T) is a T1-space and every compact subset is closed.
I know this is not a homework service but honestly I have no idea where to start. In fact, in topology I always described convergence using filters and in this question I have to prove something about sequences so I'm confused.
Can someone give me a hint?
Thanks in advance!