Closures and Infinite Products

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In summary, the conversation discusses the relationship between the closure of the product of subsets under the product topology and the box topology. The proof provided shows that the former is a subset of the latter, but it is uncertain if the reverse inclusion holds. The plan suggested is to write down the proof using open sets and see if the finiteness property of the product topology is used. If not, the proof holds for the box topology as well, but if it is used, a counterexample can be constructed.
  • #1
joeboo
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( This is from an exercise in Munkres' Topology )

Let [itex]X_\alpha[/itex] be an indexed collection of spaces, and [itex]A_\alpha \subset X_\alpha[/itex] be a collection of subsets.
Under the product topology, show that, as a subset of [itex]X = \prod_\alpha{X_\alpha}[/itex]

[tex]\overline{\prod_\alpha{A_\alpha}} = \prod_\alpha{\overline{A_\alpha}}[/tex]

This part I have no problem with. However, he then asks if this holds under the box topology on [itex]X[/itex]

Clearly ( closure being the intersection of all containing closed sets ):

[tex]\overline{\prod_\alpha{A_\alpha}} \subset \prod_\alpha{\overline{A_\alpha}}[/tex]

The reverse inclusion, on the other hand, has me in a twist.

The following is my attempt at a proof:

Let [itex]x = (x_\alpha) \in \prod_\alpha{\overline{A_\alpha}}[/itex], and [itex]U \subset X[/itex] be a neighborhood of [itex]x[/itex]. We can assume [itex]U[/itex] is a basis element for the box topology on [itex]X[/itex]
Then [itex]\exists \hspace{3} U_\alpha \subset X_\alpha[/itex] such that [itex]U = \prod_\alpha{U_\alpha}[/itex], where [itex]U_\alpha[/itex] are open.

Then we have:
[tex]x = (x_\alpha) \in U \longrightarrow x_\alpha \in U_\alpha[/tex]

Because the [itex]U_\alpha[/itex] are neighborhoods of the [itex]x_\alpha[/itex], and [itex]x_\alpha \in \overline{A_\alpha}[/itex] for all [itex]\alpha[/itex], we have:

[tex]U_\alpha \cap A_\alpha \neq \varnothing[/tex] for all [itex]\alpha[/itex]

so that:

[tex]\prod_\alpha{U_\alpha} \cap \prod_\alpha{A_\alpha} \neq \varnothing[/tex]

That, and [itex]x \in U[/itex] gives:

[tex]x \in \overline{\prod_\alpha{A_\alpha}}[/tex]

and therefore:

[tex]\prod_\alpha{\overline{A_\alpha}} \subset \overline{\prod_\alpha{A_\alpha}}[/tex]

I can't see the flaw in this proof, yet I can't somehow shake the feeling that this inclusion shouldn't hold. I know the box topology can give some funky results ( like the closure of the set of sequences with finitely many non-zero entries, or the product of continuous functions not necessarily being continuous ), so I'm a bit weary of it. That, and the way Munkres states the question gives me the suspicion it's a trick question.
So, is there a flaw with my proof? Or am I just being paranoid ( which I often tend to be )

Thanks in advance for any comments, and also for putting up with my anal-latex exactness.
 
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  • #2
joeboo said:
This part I have no problem with.
I have no proof in mind, but a plan. Write it down (I would do it with open sets instead), and find out where you used the finiteness property of the product topology. Either you didn't use it, in which case the proof holds for the box topology, too, or you see how to construct a counterexample, namely such that this step of the proof is hurt.
 

1. What is a closure in mathematics?

A closure in mathematics refers to the process of taking a set of numbers or points and combining them with all possible limits or boundary points. This results in a new set that includes the original set as well as any points that can be reached by taking limits of sequences from the original set.

2. How are closures used in real analysis?

In real analysis, closures are used to define the closure of a set, which is the smallest closed set that contains all the points in the original set. This is important for understanding the behavior of functions and sequences on a given set.

3. What is an infinite product in mathematics?

An infinite product is a mathematical expression that involves multiplying an infinite number of terms together. These products are often used in areas such as complex analysis, number theory, and calculus to represent functions and solve equations.

4. How are infinite products related to infinite series?

Infinite products and infinite series are closely related, as they both involve an infinite number of terms. However, infinite products are typically used to represent multiplicative operations, while infinite series represent additive operations.

5. What is the convergence of an infinite product?

The convergence of an infinite product refers to whether or not the product of an infinite number of terms approaches a finite value as the number of terms increases. This is an important concept in mathematics, as it determines the validity and usefulness of using infinite products in calculations and proofs.

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