Open sets and closed sets in product topology

[complexnumber]

Homework Statement

Let (X_a, \tau_a), a \in A be topological spaces, and let \displaystyle X = \prod_{a \in A} X_a.

Homework Equations

1. Prove that the projection maps p_a : X \to X_a are open maps.

2. Let S_a \subseteq X_a and let \displaystyle S = \prod_{a \in A} S_a \subseteq \prod_{a \in A} X_a. Prove that S is closed iff each S_a \subseteq X_a is closed.

3. Let T_a \subseteq X_a, prove that \displaystyle \overline{\prod_{a \in A} T_a} = \prod_{a \in A} \overline{T_a}.

4. If \abs{A} \leq \abs{\mathbb{N}} and each X_a is separable, prove that X is separable.

The Attempt at a Solution

I don't know how to prove open/closed set problems in product topology. Can someone please give me some hint as to how I should approach these proofs? Some hints on each question will be even better.

1. This means that any open subset of the product space X remains open when projected down to the X_\alpha.

Is it because the production topology \tau for X is the weakest topology with regard to \{ p_a :X \to X_a | a \in A \}?

2.

3.

4. X_a is separable means there is a countable subset S_a \subseteq X_a such that \overline{S_a} = X_a.

 

[VeeEight]

Write out what an open set in the product topology looks like: http://en.wikipedia.org/wiki/Product_topology#Definition (the second paragraph).

Based on this, where does the projection map send an arbitrary open set?

For (4), use (3): take the product of the S_a and show that it's closure is X.

 

[complexnumber]

Write out what an open set in the product topology looks like: http://en.wikipedia.org/wiki/Product_topology#Definition (the second paragraph).

Based on this, where does the projection map send an arbitrary open set?

For (4), use (3): take the product of the S_a and show that it's closure is X.

Thank you very much for your reply. Here are my answers so far based on your suggestions.

1. The product topology \tau is generated from base \mathfrak{B} consisting of product sets \displaystyle \prod_{a \in A} U_a where only finitely many factors are not X_a and the remaining factors are open sets in X_a. Therefore the project p_a projects an open set S \subseteq X to either X_a or some open subset S_a \subset X_a.

2.

3.

4. X_a is separable means there is a countable subset S_a \subseteq X_a such that \overline{S_a} = X_a. Using previous result, we have
<br /> \begin{align*}<br /> \prod_{a \in A} \overline{S_a} = \prod_{a \in A} X_a = \overline{\prod_{a \in A} S_a} = X<br /> \end{align*}<br />
Since S_a is countable and |A| \leq |\mathbb{N}|, the cartesian product \displaystyle \prod_{a \in A} S_a is countable. Hence X is separable.

***This could be wrong, if |A| = |\mathbb{N}|, then the cartesian product does not have to be countable. So what is the set separable? Should the question say If |A| &lt; |\mathbb{N}| and each |X_a| is separable, prove that |X| is separable?