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

(a) $\displaystyle \{f \in \prod_{n \in \omega} X_n | f(10)=0 \}$

(b) $\displaystyle \{f \in \prod_{n \in \omega} X_n | \text{ }\exists n \in \omega \text{ }f(n)=0 \}$

(c) $\displaystyle \{f \in \prod_{n \in \omega} X_n | \text{ }\forall n \in \omega \text{ }f(n)=0 \Rightarrow f(n+1)=1 \}$

(d) $\displaystyle \{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n)=0 \}|=5 \}$

(e)$\displaystyle \{f \in \prod_{n \in \omega} X_n | \text{ }|\{ n \in \omega | f(n)=0 \}|\leq5 \}$

Recall that $\displaystyle \omega=\mathbb{N} \cup \{ 0 \}$