Recall that the closure of a set in a topological space is defined to be the intersection of all closed sets containing it.

: By definition

but since for every we have , we immediately find

is closed

: Recall that the intersection of any number of closed sets is closed, so the closure is itself closed.

, , and

: If is any closed set, then

: First write down the definition:

then apply DeMorgan's law to get
but for every such pair , , we have that is a closed set containing . Conversely, every closed set containing is obtained from such a pair -- just take to be the pair. Thus

:

but for every such pair , , we have that is a closed set containing . However, some closed sets may not arise in this way, so we do not have equality. Thus

so we have

where is the set of all limit points of

: Let be a limit point of , and let be a closed set containing . If is not in , then is an open set containing but not meeting , which implies that does not meet , which contradicts the fact that was a limit point of . Conversely, suppose that is not a limit point of , and that is not in . Then there is some open neighborhood of which does not meet . But then is a closed set containing but not containing , so .