Math 104 Assignment 3 common mistakes

• On problem 3.A.:
  • Many people said every $x \in E^\circ$ is an interior point of $E$ so $E^\circ$ is open, but one must show in fact that $x \in E^\circ$ is an interior point of $E^\circ$.
  • Many people showed that $E^\circ$ contains all of its own interior points and concluded that it was open, but one must show that $E^\circ$ contains only its own interior points and nothing else. (In fact, any set will contain all of its own interior points, whether it is open or not; this is just the statement that in general $S^\circ\subseteq S$.)
• There was some general confusion about limit points, interior points, open sets, and closed sets. (For example, some people assumed that any set which is not closed is open, while some assumed that any set which is not open is closed; some assumed that interior points of $E$ were limit points of $E$ (this is true in $\R^d$ but not in general); and so on.)