If X is space and A is a non-empty proper open subset that is both open and closed, then is the set B = X - A both closed and open simply because the definition of a closed set is that its complement is open?
I understand it if I just follow the definition, but intuitively it doesn't make much sense, since it doesn't mention the openness/closeness of X and I don't see how the closed sets fit into the definition of a topology.
This question actually came up when I read the definition of a connected space, that is:
I can use the line of reason above to understand the definition of a connected space (i.e. the complement of B is A, which is both open and closed, hence B is both closed and open, hence A and B constitute a separation of X and X is not connected), but other than that it doesn't provide much more information.
Also, I'm new to topology, so everytime I need to determine whether a certain set is closed or open, I have the habit of thinking of the "Venn diagram", and the notion of an open set in $\Re^2$ (i.e. whether I can draw a ball around every the point of the set that fits inside the set). Clearly this doesn't apply in lots of cases. Is there a better way to think about these concepts?