I just got back from my exam and these questions' solutions eluded me, it would be great to use the rest of my evening figuring these out...
Q1: Find an open covering of the set $(0,1) \subset \mathbb{R}$, say $G =\{U_\alpha\}_{\alpha \in A}$, (where $A$ is some indexing set) such that $G$ has no finite subcover.
Q2: Let $f: [0,1] \to [0,\infty) $ be a continuous function. Let there be some $c\in [0,1]$ such that $f(c)$ is non-zero. Prove that there exists an $\epsilon \gt 0$ such that the set:
$X_1=\{\ x\in[0,1]\ | \ f(x)\gt\epsilon\ \}$
is non-empty, and open.