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# Union and Intersection of Empty Collection

 Definition/Summary A Topological Space can be defined as a non empty set X along with a class of sets, called a topology on X which is closed under (1) arbitrary unions (2) finite intersections. It can be assumed that this will always include X and p (the empty set), but understanding the reason for this can be quite difficult.

 Equations Let X be the universe P be the empty collection Q be the collection of compliments of P p the empty set U = Union I = Intersection C = Compliment Briefly, p is included as the union of the empty collection and X is included as the intersection of the empty collection. It's the latter of these that causes conceptual difficulties. How can the intersection of a collection with nothing in it be the whole universe? If we first accept that as a simple accumulation of elements it's easy to grasp that U(P) = p. Now we can use De Morgan's laws to calculate I(P). Normally when we deal with these laws we have a non empty collection of sets. So, the collection of compliments is also a non empty collection. In our case here though, there are no sets to have compliments of, so there are no compliments, so Q = P and thus U(Q) = U(P) = p. Now, by De Morgan: The compliment of the union of a collection = The intersection of the collection of compliments. So, C(U(P)) = I(Q) But, Q = P, so I(Q) = I(P) Also, C(U(P)) = C(p) = X Thus I(P) = X

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