Math 104 Assignment 9 common mistakes

• In 1A, some people showed that for some particular cover of $\set{x}$ by disjoint open sets that $\set{x}$ was a subset of only one of those sets. To argue that $\set{x}$ is connected, though, this must be true for all such covers.
• In 1C, to prove reflexivity, you must show that there is a connected subset of $A$ containing $x$ (e.g., using 1A). Many people merely said something like "Let $A \subseteq M$ be connected with $x \in A$; then $x \sim x$."
• Also on 1C, when proving transitivity, many people went immediately from $x \sim y$ and $y \sim z$ to "there is a connected set $H \subseteq A$ with $x, y, z \in H$. Instead one only has that there are two connected sets $H_1, H_2 \subseteq A$ with $x, y \in H_1$ and $y, z \in H_2$; one can then use 1B to argue $H_1\cup H_2$ is indeed connected.
• Many people used 1B to prove 2B. There is a subtlety here in that properties which hold for finite unions do not necessarily hold for infinite unions (e.g., closedness or compactness). It turns out that the infinite version of 1B is true, but this requires some care. A better approach would be to show directly that any two points in $M$ related by $\sim_p$ and therefore by $\sim$. Then $M$ consists of a single $\sim$ equivalence class, and so is connected.
• On 2C, many people assumed (incorrectly) that image of an open set under a continuous function is open; it is the opposite that is true.
• In 2C, many people assumed without proof that open balls in $\R^n$ are path connected. This is true, but not obvious. The grader decided to let this one slide, though.
• In 4, when showing \[\lim_{h\to0} \frac{f(h)-f(0)}{h-0} = 0,\] many people ignored the case where $h \in \R\setminus\Q$.