Thread: subsets/open sets

For any set S a subset of R, let S^o denote the union of all the open sets contained in S
A) prove that S^o is an open set
B) prove that S^o is the largest open set contained in S. That is, show that S^o a subset of S, and if U is any open set contained in S, then U a subset of S^o
C) prove that S^o = int S

Originally Posted by luckyc1423

For any set S a subset of R, let S^o denote the union of all the open sets contained in S
A) prove that S^o is an open set
B) prove that S^o is the largest open set contained in S. That is, show that S^o a subset of S, and if U is any open set contained in S, then U a subset of S^o
C) prove that S^o = int S

No sure what definition of ‘open’ you are using. This one is standard: A set S is open iff for each x is S there is a open set O_x which contains x and is a subset of S.

Do you see how part A) results directly from that definition.

For part B), any open set of S is a subset of S^o.

For part C), a point t is in int(S) iff there is a open set O_t which contains t and is a subset of S.