Its great when something makes you smile and this totally made me smile today. The idea that a set can be both open and closed at the same time. Its a very simple proof but since its circular, it leaves you spinning till, well till I had to have a laugh.
First consider the following definition: The set A is considered closed if A = cl(A), or the closure of A. A is considered open if Ac, or the compliment of A, is closed.
To the fun bit: Now all this takes place inside set X, nothing exists outside of it, so X is everything. The opposite, or compliment of X, Xc, would be ∅, or the empty set. The theorem states X and ∅ are both open and closed. How is this?
First lets start with X: X is closed if X = cl(X), which is true since 1) X is a subset of cl(X) and 2) since X is everything, cl(X) is a subset of X, therefore X=cl(X). Now consider ∅, the closure of ∅ is ∅, so ∅ is closed. The compliment of X would be ∅, and since ∅ is closed, X is open. Using the previous result, since X is closed, its compliment, ∅, would be open.