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## Main Question or Discussion Point

For each [tex]n \in \omega[/tex], let [tex]X_n[/tex] be the set [tex]\{0, 1\}[/tex], and let [tex]\tau_n[/tex] be the discrete topology on [tex]X_n[/tex]. For each of the following subsets of [tex]\prod_{n \in \omega} X_n[/tex], say whether it is open or closed (or neither or both) in the product topology.

(a) [tex]\{f \in \prod_{n \in \omega} X_n | f(10) = 0 \}[/tex]

(b) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }\exists n \in \omega \text{ }f(n) = 0 \}[/tex]

(c) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }\forall n \in \omega \text{ }f(n) = 0 \Rightarrow f(n + 1) = 1 \}[/tex]

(d) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}| = 5 \}[/tex]

(e)[tex]\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}|\leq5 \}[/tex]

(a) [tex]\{f \in \prod_{n \in \omega} X_n | f(10) = 0 \}[/tex]

(b) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }\exists n \in \omega \text{ }f(n) = 0 \}[/tex]

(c) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }\forall n \in \omega \text{ }f(n) = 0 \Rightarrow f(n + 1) = 1 \}[/tex]

(d) [tex]\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}| = 5 \}[/tex]

(e)[tex]\{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n) = 0 \}|\leq5 \}[/tex]