Ian Charlesworth

Let $(\mathcal{M}, d)$ be a metric space.

1. If $E \subseteq M$ is not compact, then no open cover of $E$ has a finite subcover.
   1. True
   2. False
2. If $E \subseteq M$ is not compact, then no infinite open cover of $E$ has a finite subcover.
   1. True
   2. False
3. Suppose $E \subset M$, and let $\mathcal{U}$ be an open cover of $E$. Then $\mathcal{U}$ has a supercover which admits a finite subcover, i.e., there is an open cover $\mathcal{V}$ with $\mathcal{U} \subseteq \mathcal{V}$ so that $\mathcal{V}$ has a finite subcover.
   1. True
   2. True iff $E$ is compact
   3. False
4. Any set with no limit points is closed.
   1. True
   2. False
5. Any bounded set with no limit points is compact.
   1. True
   2. False