Infinite index set in product topology

[pieterb]

Homework Statement

Let

$$Y := \prod_{i \in I} X_i$$

Now assume $$U_i \subset X_i$$ to be open.

If we take i to be infinite, $$\prod_{i \in I} X_i$$ cannot be open. Why?

Homework Equations

The Attempt at a Solution

I can't quite get my head around how to approach this problem. A part of me says that the assumption that $$U = pr^-1(U_i) intersection ..$$ does not produce an open set. (because the theorem for topological spaces says that only a finite intersection of open spaces is open).

How would one go about proving this rigorously?

Thanks in advance.

 

[Dick]

Show there is no open set U in the basis of the topology such that U is a subset of Y.

 

[pieterb]

Have been sitting on this while doing the dishes, however no new insights. Am I in the right direction by thinking that the fact that U is defined by infinite intersections no longer guarantees it to be open?

 

[Dick]

Have been sitting on this while doing the dishes, however no new insights. Am I in the right direction by thinking that the fact that U is defined by infinite intersections no longer guarantees it to be open?

The question isn't about intersections. The big Pi symbol doesn't mean intersection. What are the basis sets that define your product topology? Look it up.

 

[pieterb]

According to my textbook, the following gives B, the collection of subsets that forms the basis.

$$\cap_{a i}^{n} pr^{-1}_{a i}(U_{a i}) = \left\{(x_{\alpha})_{\alpha \in A} \in Y | x_{a i} \in U_{a i} for all i = 1, ... , n\right\}$$

Given $$a1,..,an \in A$$ and $$U_{a i}$$ open in $$X_{a i}$$

I don't see why this cannot be extended towards infinity.

 

[Dick]

According to my textbook, the following gives B, the collection of subsets that forms the basis.

$$\cap_{a i}^{n} pr^{-1}_{a i}(U_{a i}) = \left\{(x_{\alpha})_{\alpha \in A} \in Y | x_{a i} \in U_{a i} for all i = 1, ... , n\right\}$$

Given $$a1,..,an \in A$$ and $$U_{a i}$$ open in $$X_{a i}$$

I don't see why this cannot be extended towards infinity.

Here's what is maybe a clearer way to describe that. A set U is in the basis if it's given by
$$U = \prod_{i \in I} U_i$$
where U_i is a open subset of X_i and U_i=X_i for ALL BUT A FINITE NUMBER of indices i. Your set is
$$\prod_{i \in I} U_i$$
where U_i is a proper open subset of X_i. (At least that's what I'm guessing from the problem statement). So U_i isn't equal to X_i for ANY i. And, yes, putting it that way you can say the problem is the need to do an infinite number of intersections.

 

[pieterb]

Wow, I was thinking in completely the wrong direction. Thanks so much for your assistance!

 

[Dick]

Wow, I was thinking in completely the wrong direction. Thanks so much for your assistance!

So you see the problem? An infinite product of proper open subsets is too small to have any basis element contained in it.

 

[pieterb]

I see the problem, currently trying formulate it in a neat way. Any tips?

 

[Dick]

I see the problem, currently trying formulate it in a neat way. Any tips?

Like said before. Take any U in the basis and show it cannot be contained in your set.