For any set S which is a subset of R (real numbers), let S' denote the intersection of all the closed sets con

For any set S which is a subset of R (real numbers), let S' denote the intersection of all the closed sets containing S.


a) Prove that S' is closed set.


b) Prove that S' is the smallest closed set containing S,i.e. show that S%26lt;=S', and if c is any closed set containing S, then show that S'%26lt;=C.

For any set S which is a subset of R (real numbers), let S' denote the intersection of all the closed sets con
it helps to know what the context of your class is, since for example there are different ways to define what it means for a set to be closed. I'm guessing that the right definition for you is that a set is closed if it's the complement of an open set, and an open set is one where every point has an open neighborhood contained in the set.





a) it's fairly easy to see that the complement of S' is equal to the union of all complements of closed sets containing S. it's also fairly easy to see that an arbitrary union of open sets is again an open set (it may even be a theorem that your class has already proven). therefore S' is the complement of an open set, and is closed.





b) since S' is the intersection of *all* closed sets, any point that's not in C can't be in S'. therefore S' is a subset of C.

blazing star