Find Disjoint Subspaces A,B to Connected Space

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In summary, the conversation discusses finding disjoint, non-empty, disconnected subspaces A and B that are open and when their union is taken, results in a connected set. The idea of using A=\mathbb{Z} and B=\mathbb{R} \backslash \mathbb{Z} is proposed, but the open condition of A and B is questioned. Eventually, it is determined that the original problem may be impossible to solve if the term "disconnected" is interpreted as "separated".
  • #1
latentcorpse
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I've been asked to find disjoint, non-empty, disconnected subspaces [itex]A,B \subset \mathbb{R}[/itex] such that [itex]A \cup B[/itex] is connected.

My problem is in that because the A and B are open and disjoint, when i take the union i keep getting one point omitted which prevents the union from being connected.

i was wondering about [itex]A=\mathbb{Z}[/itex] and [itex]B=\mathbb{R} \backslash \mathbb{Z}[/itex]. These are disjoint, non-empty and open subsets of the real line and when u take their union you get [itex]\mathbb{R}[/itex] which is connected. I'm not sure about the disconnectedness of A and B though...
 
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  • #2
Z is NOT open. And R is not the disjoint union of two open sets (disconnected or not). The first statement you made of the problem says nothing about A and B being open.
 
  • #3
sry. that was a big mistake. i think I've got it now though, thanks!
 
  • #4
latentcorpse said:
I've been asked to find disjoint, non-empty, disconnected subspaces [itex]A,B \subset \mathbb{R}[/itex] such that [itex]A \cup B[/itex] is connected.
Did the problem really say "disconnected"? What does that mean? If it just means "disjoint", you don't need to say it. My first thought was to interpret it as "separated" but then this problem is impossible.
 

Related to Find Disjoint Subspaces A,B to Connected Space

1. What is the definition of a connected space?

A connected space is a topological space in which there are no two disjoint non-empty open sets. This means that any two points in the space can be joined by a path without leaving the space.

2. Why is it important to find disjoint subspaces in a connected space?

Finding disjoint subspaces in a connected space allows us to better understand the structure and properties of the space. It can also help us to simplify complex systems by breaking them down into smaller, more manageable subspaces.

3. How do you determine if two subspaces are disjoint?

To determine if two subspaces, A and B, are disjoint, we need to check if their intersection is empty. This means that there are no points that are in both A and B. If the intersection is empty, then A and B are disjoint.

4. Can a connected space have more than two disjoint subspaces?

Yes, a connected space can have more than two disjoint subspaces. In fact, a connected space can have an infinite number of disjoint subspaces. This is because a connected space is not limited to just two subspaces, but can have multiple subspaces that are not connected to each other.

5. How can the concept of disjoint subspaces be applied in real-world situations?

The concept of disjoint subspaces has many applications in real-world situations. For example, it can be used in network analysis to identify separate components within a larger network. It can also be applied in data analysis to identify distinct clusters or groups within a dataset. Additionally, in physics, the concept of disjoint subspaces is used to study the behavior of particles in quantum mechanics.

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