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complexnumber
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Homework Statement
Let [tex](X_a, \tau_a), a \in A[/tex] be topological spaces, and let [tex]\displaystyle X = \prod_{a \in A} X_a[/tex].
Homework Equations
1. Prove that the projection maps [tex]p_a : X \to X_a[/tex] are open maps.
2. Let [tex]S_a \subseteq X_a[/tex] and let [tex]\displaystyle S = \prod_{a \in A} S_a \subseteq \prod_{a \in A} X_a[/tex]. Prove that [tex]S[/tex] is closed iff each [tex]S_a \subseteq X_a[/tex] is closed.
3. Let [tex]T_a \subseteq X_a[/tex], prove that [tex]\displaystyle \overline{\prod_{a \in A} T_a} = \prod_{a \in A} \overline{T_a}[/tex].
4. If [tex]\abs{A} \leq \abs{\mathbb{N}}[/tex] and each [tex]X_a[/tex] is separable, prove that [tex]X[/tex] 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 [tex]X[/tex] remains open when projected down to the [tex]X_\alpha[/tex].
Is it because the production topology [tex]\tau[/tex] for [tex]X[/tex] is the weakest topology with regard to [tex]\{ p_a :X \to X_a | a \in A \}[/tex]?
2.
3.
4. [tex]X_a[/tex] is separable means there is a countable subset [tex]S_a \subseteq X_a[/tex] such that [tex]\overline{S_a} = X_a[/tex].