we're going to prove two results about closed sets first we'll prove that if we take the union of a finite collection of closed sets we will still have a closed set the union of a finite number of closed sets is a closed set then we will prove that an arbitrary intersection of closed sets is also a closed set so if you have any number of closed sets whatsoever the intersection of all of them is also closed both of these proofs will be short but to make them easy we need to cover a couple preliminary things first I hope you recall de Morgan's laws which tell us how compliments work with unions and intersections of sets the complement of a union of sets is the intersection of the sets complements and similarly for the complement of an intersection for our proofs we will use the generalized form of De Morgan's laws it's also important that we recall the definition of a close set and I'll leave a link in the description to my lesson introducing closed sets a definition of a closed set is that it's closed if its complement is open there is another equivalent definition but this is indeed a definition of closed sets a set is closed if it's complement is open and this will allow us to use de Morgan's laws to connect our closed sets to open sets and then we can use previously proven results about open sets and of course these are the results about open sets that we will use I'll leave a link in the description to my lesson where we prove these results we will use the fact that an arbitrary Union of open sets is also an open set and we will use the result that a finite intersection of open sets is also an open set with all that said we can make short work of these proofs first here is our proof that a finite Union of closed sets is also so closed we've got this finite collection of closed sets U1 U2 all the way up through u n so each of these closed sets say UK certainly has a complement which is open by definition the complement of a closed set has to be an open set then we could imagine taking the intersection of this finite collection of open sets take the complement of every one of these closed sets and we'll have a finite collection of open sets and we could take their intersection and we've already proven that a finite intersection of open sets is also open so this finite intersection of open sets of course is open now if we take the complement of this finite intersection we will get a closed set because by definition the complement of an open set is closed so we're going to take the complement of this open set to get a closed set but as it turns out by De Morgan's laws the complement of this finite intersection is exactly the finite Union that we're trying to prove our result about and that is because when we take the complement of this finite intersection each intersection becomes a union and each complement of the closed sets right each UK complement becomes the original closed sets UK and again since this is the complement of an open set it has to be closed so we've proven that this finite Union of our closed sets is closed once more the fact that the complement of this open set is closed is just by definition but the fact that the complement of this open set equals this Union of closed sets is a result of De Morgan's law and the proof for an arbitrary intersection of closed sets is very similar we could take the complement of any one of these closed sets and certainly say U Alpha complement it would have to be open because the complement of a closed set is open but then we could consider the arbitrary Union of all of these open sets the union of every U Alpha complement by a previous proof we know that an arbitrary Union of open sets has to be open then if we take the complement of this arbitrary Union we will by definition get a closed set but again by De Morgan's law we'll see that the complement of this Union is exactly the intersection we're looking for because if we take the complement of the Union of each U Alpha complement then the unions become intersections and each U Alpha complement becomes its complement which is just U Alpha the original closed sets the that we were talking about and so we have that this arbitrary intersection of closed sets has to be closed because we found that it actually equals the complement of an open set and those are the proofs so if you have an arbitrary Union of open sets you still have an open set if you have an arbitrary intersection of closed sets then you still have a closed set if you have a finite intersection of open sets then you still have an open set and if you have a finite Union of closed sets then you still have a closed set thanks for watching let me know in the comments if you have any questions [Music] sink into the stomachs that plummeted at the accident shot off all my habits to addicts ripped by a catapult pulled apart the patterns in man that stood as a manifold man of many pains with a number that he had to call calculate the damage of himself and what's collateral being told he's not enough until he swallows Adderall oh