Proof: Topology of subsets on a Cartesian product

In summary: For this, you use the fact that both Tx and Ty are topologies, so ##\bigcap_{k=1}^n E_k## is the cartesian product of...what? In summary, The question asks if T = { A × B : A\inTx, B\inTy } is a topology on X × Y given that Tx and Ty are topologies on X and Y respectively. To prove this, the conditions (i) must be satisfied which states that (∅, ∅)∈T and (X × Y)∈T. To prove this, it is necessary to show that ∅∈A and ∅∈B, which may not always be true since the empty set
  • #1
Colossus91
1
0
Homework Statement

Let Tx and Ty be topologies on X and Y, respectively. Is T = { A × B : A[itex]\in[/itex]Tx, B[itex]\in[/itex]Ty } a topology on X × Y?

The attempt at a solution

I know that in order to prove T is a topology on X × Y I need to prove:
i. (∅, ∅)[itex]\in[/itex]T and (X × Y)[itex]\in[/itex]T
ii. T is closed under finite intersections
iii. T is closed under arbitrary unions

In order to prove (i) I would have to prove that ∅[itex]\in[/itex]A and ∅[itex]\in[/itex]B. I think this is true because the empty set is in all sets.
I'm not sure how to approach proving that X[itex]\in[/itex]A as even though A[itex]\in[/itex]Tx, this implies that A[itex]\in[/itex]X or A is X. I'm not sure how continue from here. Same with Y[itex]\in[/itex]B.

For ii. I think that since Tx and Ty are topologies themselves, they are closed under finite intersections, and since A[itex]\in[/itex]Tx and B[itex]\in[/itex]Ty then A and B are also closed under finite intersections, thus T is closed under finite intersections. I have to go more into detail with this but I just want to make sure if this is the right idea.

I think iii. could also be proved with a similar argument to the one used to prove ii.
 
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  • #2
For (i), your topology T is the set of all open sets A x B such that A is an element of T_x, and B is an element of T_y. X is an element of T_x, as it is required to be one by the same rules we a re trying to prove, as well as the empty set, and vice versa for Y being an element on T_y. Use this fact to show that X x Y is in your topology T.
 
  • #3
Colossus91 said:
i. (∅, ∅)[itex]\in[/itex]T and (X × Y)[itex]\in[/itex]T
This one should say ##\emptyset\in T## and ##X\times Y\in T##.

Colossus91 said:
In order to prove (i) I would have to prove that ∅[itex]\in[/itex]A and ∅[itex]\in[/itex]B. I think this is true because the empty set is in all sets.
It's not. It's a subset of all sets, but most sets don't have it as a member.

Colossus91 said:
I'm not sure how to approach proving that X[itex]\in[/itex]A as even though A[itex]\in[/itex]Tx, this implies that A[itex]\in[/itex]X or A is X. I'm not sure how continue from here. Same with Y[itex]\in[/itex]B.
It doesn't make sense to try to prove that X is a member of A when you haven't specified what A is.

##A\in T_x## doesn't imply what you say it implies. It just means that A is an open subset of X.

Note that the definition of T says that T consists of of all cartesian products of two open subsets of X and Y, such that the first set is a subset of X and the second a subset of Y. To check if ##\emptyset\in T##, you should ask yourself if ∅ can be expressed as a cartesian product at all. X×Y is obviously a cartesian product, so to prove that X×Y is in T, you only have to prove...what?

Colossus91 said:
For ii. I think that since Tx and Ty are topologies themselves, they are closed under finite intersections, and since A[itex]\in[/itex]Tx and B[itex]\in[/itex]Ty then A and B are also closed under finite intersections, thus T is closed under finite intersections. I have to go more into detail with this but I just want to make sure if this is the right idea.
I don't see what A and B being closed under finite intersections have to do with anything. You need to start with a statement like "Let n be an arbitrary positive integer, and let ##E_1,\dots,E_n## be arbitrary members of T". Then you prove that ##\bigcap_{k=1}^n E_k\in T##.
 

1. What is a Cartesian product?

A Cartesian product, also known as a cross product, is a mathematical operation that combines two sets to form a new set. The resulting set contains all possible ordered pairs of elements from the original sets.

2. What is a topology?

A topology is a mathematical structure that studies the properties of a set that remain unchanged through continuous deformations, such as stretching or bending. It is used to define the concept of continuity and convergence in mathematics.

3. What is the topology of subsets on a Cartesian product?

The topology of subsets on a Cartesian product is a mathematical framework for studying the properties of subsets of a Cartesian product. It defines the concept of open and closed sets, as well as continuity and convergence, in relation to the Cartesian product.

4. How is the topology of subsets on a Cartesian product used in real-world applications?

The topology of subsets on a Cartesian product has many practical applications, such as in computer science, engineering, and physics. It can be used to study the behavior of complex systems, model physical phenomena, and analyze data in various fields.

5. What is the importance of understanding the topology of subsets on a Cartesian product?

Understanding the topology of subsets on a Cartesian product is important because it provides a powerful tool for studying and analyzing mathematical and real-world systems. It allows for the identification of important properties and relationships between sets, and can be used to solve complex problems in various fields of study.

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