Question on the closure of an intersection of sets

[theverymooon1]

Im studying metric space topology and the closure of sets. I'm having trouble wrapping my head around this statement. 'The closure of an intersection of sets is always a subset of (but need not be equal to) the intersection of the closures of the sets.' What I don't understand is the 'but need not be equal' part. How are they not equal? Or in symbol form, why is Closure (X intersection Y) not equal to Closure(X) intersection Closure(Y)? (I'm pretty sure I understand the concept of closure and limit points..) Any insight into this matter would be much appreciated!!

 

[stapel]

What I don't understand is the 'but need not be equal' part. How are they not equal?

Try checking out the example at the end of this article. :wink:

 

[HallsofIvy]

Let A= (0, 1) and B= (1, 2). The intersection of the two sets is empty and since the empty set is closed, it closure is the empty set. But the closure of the two sets is [0, 1] and [1, 2] and the intersection of those sets is {1}.