Showing the connectedness of a product topology

In summary: This is done by showing that (C u E) and (D u F) form a separation between X and Y, which cannot occur. This ultimately proves that X × Y − A × B is connected. In summary, to prove that X × Y − A × B is connected, you need to show that there is no separation between X × Y − A × B, which can be done by assuming a separation (U u V) = (X x Y) - (A x B), and then showing that this contradicts the fact that X × Y is connected.
  • #1
trap101
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Let A be a proper subset of X, and let B be a proper subset of Y .
If X and Y are connected, show that X × Y − A × B is connected.

Attempt: Proven before in my book, I know that since X and Y are each connected that X x Y is also connected. Keeping that fact in mind.

Pf: Assume (X x Y) - (A x B) has a separation.

--> (U u V) = (X x Y) - (A x B) where U and V are open disjoint nonempty subsets.

U and V can each be written in terms of their basis elements of the form (C x D) and (E x F) where C,E [itex]\subset[/itex] X and D,F [itex]\subset[/itex] V

therefore: (U u V) = (C x D) u (E x F)
= (C u E) x (D u F)
--> (C u E) [itex]\subset[/itex] X and (D u F)[itex]\subset[/itex]Y

Therefore I formed a separation between X and Y, but this cannot occur because we know
(X x Y) is connected --> Contradiction.


Now I feel something is a little messed up in the proof, but the idea is that I want to show that
(X x Y) has a separation which contradicts the fact I know about it.
Am I even close in my idea?
 
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  • #2
Yes, you are on the right track. To prove that X × Y − A × B is connected, you need to show that there is no separation between X × Y − A × B. You can do this by assuming that there is a separation (U u V) = (X x Y) - (A x B) where U and V are open disjoint nonempty subsets, and then showing that this contradicts the fact that X × Y is connected.
 

1. What is a product topology?

A product topology is a type of topology that is defined on the Cartesian product of two or more topological spaces. It is used to describe the topology of a space that is composed of the combination of two or more subspaces.

2. How is the connectedness of a product topology determined?

The connectedness of a product topology is determined by the connectedness of its individual subspaces. In order for the product topology to be connected, all of its subspaces must also be connected.

3. Can a product topology be disconnected?

Yes, a product topology can be disconnected if any of its individual subspaces are disconnected. This means that the product topology can be split into two or more disconnected components.

4. How does the connectedness of a product topology affect its properties?

The connectedness of a product topology can affect its properties in various ways. For example, a connected product topology may have different properties than a disconnected product topology, such as different compactness or completeness properties.

5. What is the importance of showing the connectedness of a product topology?

Showing the connectedness of a product topology is important because it helps to understand the topological structure of the space. It also allows for the application of various properties and theorems that are specific to connected spaces, which can aid in further analysis and understanding of the topology.

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