My friend and I are still stuck on:

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

Background:

If it helps you can think of $\displaystyle \prod_{n\in \omega} X_n$ as $\displaystyle \prod_{n=0}^{\infty} X_n$. We define $\displaystyle \prod_{n=0}^{\infty}X_n$ to be the set of all functions $\displaystyle f: \mathbb{N} \to \{ 0 , 1\}$ that satisfies $\displaystyle f(n) \in \{ 0 , 1\}$.

There's a nice graphical representation of the product topology on $\displaystyle Y^X$ (i.e. the product of the space $\displaystyle Y$ $\displaystyle |X|$ times). Namely, if we draw $\displaystyle X$ as an "$\displaystyle x-$axis" and $\displaystyle Y$ as a "$\displaystyle y-$axis", then elements in $\displaystyle X^Y$ are "graphs of functions" in the $\displaystyle X-Y$ "plane". An open nbhd of an element $\displaystyle f$ is the set of all functions $\displaystyle g$ whose graphs are close to the graph of $\displaystyle f$ at finitely points. We get different nbhds by varying the closeness to $\displaystyle f$ and/or the set of finite points.

In our case the product space is $\displaystyle 2^\omega=2^\mathbb{N}$, whose "plane" looks like two copies of the naturals $\displaystyle \mathbb{N}$. In other words, if you were to imagine this as a 'subset' of $\displaystyle \mathbb{R}^2$, it's just the set $\displaystyle \{(n,i) : n \in \mathbb{N}, i \in \{0,1\}\} $.

Progress:

So remember that open sets in the infinite product topology is really just having all but finitely many the whole space and the rest are open. Since the individual factors are discrete, you only need to check that all but finitely many are the whole space.

e.g. in (a) the 10th coordinate has a specific value, but all other coordinates can be whatever, so this is certain open.