Open Subsets of a Union: A Conjecture

In summary, the conversation discusses a conjecture about the openness of subsets in a union of subsets of a given set. It is pointed out that the conjecture is false, as there exist counterexamples in non-discrete topologies. Examples are provided to illustrate this point.
  • #1
pivoxa15
2,255
1

Homework Statement


Conjecture: If K=a union of subsets of G with K open then each subset in the union is open

The Attempt at a Solution


Can't really see the proof. In fact it's false as any non discrete topology have open sets which are a union of subsets whch may not be open.
 
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  • #2
How do you expect to see the proof if you already know that the statement is false?!
 
  • #3
Or consider the classic example where one takes the reunion of the non-opens sets [1/n,+infty) and get the open sets (0,+infty)
 
  • #4
You can practically create a counterexample for any topology except that of the discrete topology.
 
  • #5
quasar987 said:
Or consider the classic example where one takes the reunion of the non-opens sets [1/n,+infty) and get the open sets (0,+infty)

Why not just take the union of 0 and (-1,1). We get the open set (-1,1) but the point 0 is closed.
 
  • #6
morphism said:
How do you expect to see the proof if you already know that the statement is false?!

After I created this thread, I realized the conjecture was false.
 

1. What is an open subset?

An open subset is a subset of a set that contains all of its limit points. In other words, every point in an open subset has a neighborhood that is also contained in the subset.

2. Can you provide an example of an open subset of a union?

Yes, consider the union A ∪ B, where A = {1, 2, 3} and B = {2, 3, 4}. An open subset of this union could be {2, 3}, since both 2 and 3 have neighborhoods that are contained in the union.

3. What is the conjecture about open subsets of a union?

The conjecture states that for any two sets A and B, if A and B are open subsets of a union, then their union A ∪ B is also an open subset of the same union.

4. Why is this conjecture important?

This conjecture is important because it helps to understand the properties of open subsets and unions. It also has implications in topology and analysis, as open subsets are fundamental concepts in these fields.

5. Has this conjecture been proven or disproven?

This conjecture has been proven to be true. It follows from the definition of open subsets and unions, and can be easily verified using set theory principles.

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