Exploring the Relationship Between Open and Closed Sets in Topology

That implies B(S) and B(S^C) are empty sets, which means the boundary of S is empty. Therefore, if S is open and S^C is open, then the boundary of S must be empty.
  • #1
Design
62
0

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
 
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  • #2
Yes, S and S^C always have the same boundary. Show that by quoting the definition of 'boundary'.
 
  • #3
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?
 
  • #4
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


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

Sc is open means boundary of Sc is a subset of S. Since you have shown that the boundary of Sc 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.
 
  • #5
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?

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

1. What is the difference between open and closed sets in topology?

In topology, a set is considered open if all of its points are contained within the set's interior. On the other hand, a closed set is one that contains all of its limit points. This means that for every point on the boundary of a closed set, there exists a sequence of points within the set that converges to that boundary point.

2. How are open and closed sets related to each other?

In some cases, open and closed sets can be complementary. This means that if a set is open, then its complement (the set of all points not contained within the set) is closed, and vice versa. However, this is not always the case, as there are sets that are both open and closed, such as the entire space in which they are defined.

3. Can a set be neither open nor closed?

Yes, a set can be both neither open nor closed. These sets are commonly referred to as "half-open" or "half-closed" sets. They have some points that are contained in their interior but also have some points that are on their boundary and not included in the set.

4. How do open and closed sets relate to the concept of continuity in topology?

In topology, a function is considered continuous if and only if the preimage of every open set is open. This means that open sets are preserved under continuous functions. Similarly, closed sets are preserved under continuous functions, as the preimage of a closed set is also closed. This is an important concept in topology that helps to define the continuity of functions.

5. Can a set be both open and closed at the same time?

Yes, a set can be both open and closed at the same time. These sets are known as clopen sets. In topology, clopen sets are often used to define connectedness, as a set is considered connected if it is not possible to divide it into two non-empty clopen subsets.

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