What Is the Relationship Between Open Sets and Their Boundaries in Topology?

[Design]

Homework Statement

Prove that if S is open and S^c 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 S^c
S^c is open means boundary of S^c 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

 

[Dick]

Yes, S and S^C always have the same boundary. Show that by quoting the definition of 'boundary'.

 

[Design]

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?

 

[sephy]

Homework Statement

Prove that if S is open and S^c 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 S^c
S^c is open means boundary of S^c 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

I think you've pretty much done it -
S is open means the boundary of S is a subset of S^c, so the boundary is not in S.

S^c is open means boundary of S^c is a subset of S. Since you have shown that the boundary of S^c is equal to the boundary of S, this implies that the boundary of S is a subset of S, but S is open so this cannot be.

 

[Dick]

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?

You already did that in the first post if you know boundary of S=boundary of S^C.