The discussion revolves around the relationship between open sets and their boundaries in the context of topology. The original poster presents a problem requiring a proof that if a set S is open and its complement Sc is also open, then the boundary of S must be empty.
Some participants have provided insights into the definitions involved and how they relate to the problem. There is recognition of the relationship between the boundaries of S and Sc, but no explicit consensus has been reached on the proof or the explanation of why the boundary is empty.
Participants are working under the constraints of the problem statement and are attempting to clarify definitions and relationships without providing a complete solution. There is an ongoing exploration of the implications of the definitions of boundary and open sets.
Design said:Homework Statement
Prove that if S is open and Sc is open then boundary of S must be empty
The Attempt at a Solution
S is open means boundary of S is a subset of Sc
Sc is open means boundary of Sc is a subset of S (By taking complement of both sides from the definition ?)
This means that they have the same boundary?
Don't know how to proceed from here
thanks
Design said:For all r, B(r,x) intersection S = not empty and B(r,x) intersection S^C = not empty, Ok I see how i got this from definition of boundary for both of them. How do I explain the set is empty?