The Union of Two Open Sets is Open

In summary, the conversation discusses the proof that if x ∈ A1 ∪ A2, then x ∈ A1 or x ∈ A2. This is based on the definition of an open set, which states that for every point x in the set, there exists an open ball centered at x contained in the set. By looping over A1 and A2, it is shown that all points in A1 ∪ A2 have open balls contained in the union, thus proving that A1 ∪ A2 is an open set.
  • #1
G-X
21
0
Let \(\displaystyle x ∈ A1 ∪ A2\) then \(\displaystyle x ∈ A1\) or \(\displaystyle x ∈ A2\)

If \(\displaystyle x ∈ A1\), as A1 is open, there exists an r > 0 such that \(\displaystyle B(x,r) ⊂ A1⊂ A1 ∪ A2\) and thus B(x,r) is an open set.

Therefore \(\displaystyle A1 ∪ A2\) is an open set.

How does this prove that \(\displaystyle A1 ∪ A2\) is an open set. It just proved that \(\displaystyle A1 ∪ A2\) contains an open set; not that the entire set will be open? This is very similar to the statement: An open subset of R is a subset E of R such that for every x in E there exists ϵ > 0 such that Bϵ(x) is contained in E.
 
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  • #2
The point is that the argument is valid for every $x\in A_1\cup A_2$.

If $C = A_1\cup A_2$, we have proved that, for every $x\in C$, there is an open ball $B(x,r)\subset C$ (where $r>0$ depends on $x$). That is precisely the definition of an open set.
 
  • #3
G-X said:
If \(\displaystyle x ∈ A1\), as A1 is open, there exists an r > 0 such that \(\displaystyle B(x,r) ⊂ A1⊂ A1 ∪ A2\) and thus B(x,r) is an open set.

Therefore \(\displaystyle A1 ∪ A2\) is an open set.

Hi G-X, welcome to MHB!As castor28's pointed out, it's about the definition of an open set, which he effectively quoted.Additionally that proof is not entirely correct and it is incomplete.
It should be for instance:

If \(\displaystyle x ∈ A1\), as $A1$ is open, there exists an $r > 0$ such that \(\displaystyle B(x,r) ⊂ A1\) (from the definition of an open set), which implies that \(\displaystyle B(x,r)⊂ A1 ∪ A2\).
If \(\displaystyle x ∈ A2\), as $A2$ is open, there exists an $r > 0$ such that \(\displaystyle B(x,r) ⊂ A2⊂ A1 ∪ A2\).
Therefore for all \(\displaystyle x ∈ A1∪ A2\), there exists an $r > 0$ such that \(\displaystyle B(x,r) ⊂ A1 ∪ A2\).

Thus \(\displaystyle A1 ∪ A2\) is an open set.
 
  • #4
I see, I think I had the misunderstanding that something from A2 might close A1.

But I don't think that is an issue you technically need to wrap your head around.

Because the definition states: We define a set U to be open if for each point x in U there exists an open ball B centered at x contained in U.

So, essentially looping over A1, A2 - making the reference that open balls exist at each of these points then all points in A1 ∪ A2 have open balls contained in the union thus by the definition it must be open.
 
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Related to The Union of Two Open Sets is Open

1. What is the definition of "The Union of Two Open Sets is Open"?

The Union of Two Open Sets is Open refers to a mathematical concept in topology where the union of two open sets is also an open set.

2. How is the union of two open sets related to topology?

The union of two open sets is a fundamental concept in topology, as it helps define the topology of a space by determining which sets are open and which are not.

3. Why is it important to understand the union of two open sets in topology?

Understanding the union of two open sets is crucial in topology because it allows us to define and study topological spaces, which have applications in various fields such as physics, computer science, and engineering.

4. Can the union of two open sets be closed?

No, the union of two open sets cannot be closed. In topology, open sets are defined as sets that do not contain their boundary points. Therefore, the union of two open sets will also not contain any boundary points, making it an open set as well.

5. What is an example of the union of two open sets being open?

An example of the union of two open sets being open is the union of the intervals (0,2) and (3,5) on the real number line. The union, (0,2) ∪ (3,5) = (0,2,3,5), is also an open interval on the real number line, satisfying the definition of the union of two open sets being open.

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