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Math 185 - Material from previous courses

Below is a list of definitions and content from previous courses (mostly 104) which I will be using freely this term.

Open
A set $U \subseteq \C$ is called open if for every $z \in U$ there is some $r \gt 0$ so that for any $w \in \C$ with $|z - w| \lt r$ we have $w \in U$.
Closed
A set $F \subseteq \C$ is called closed if whenever $z \in \C$ is such that for every $r \gt 0$ there is $w \in F$ with $|w - z| \lt r$, we have $z \in F$. (That is, if $F$ contains all its limit points.)
A set is closed if and only if its complement is open.
Sequences
A sequence $(z_n)_n$ in $\C$ is said to converge to a limit $z \in \C$ if for every $\epsilon \gt 0$ there is some $N \in \N$ so that whenever $n \gt N$ we have $|z - z_n| \lt \epsilon$.
A sequence $(z_n)_n$ in $\C$ is said to be Cauchy if for every $\epsilon \gt 0$ there is some $N \in \N$ so that whenever $n, m \in \N$ with $n, m \gt N$ we have $|z_n - z_m| \lt \epsilon$.
Convergent sequences are Cauchy. A metric space is complete if every Cauchy sequence converges; $\R$ and $\C$ are complete.
Compact
A set $K \subseteq \C$ is compact if for every open cover $\mathscr{U}$ of $K$ (that is, every set $\mathscr{U}$ whose elements are open subsets of $\C$ such that $K \subseteq \displaystyle\bigcup_{U \in \mathscr{U}} U$), there is a finite subcover $\mathscr{V}\subseteq \mathscr{U}$ of $K$ (that is, $\mathscr{V}$ remains an open cover of $K$, but consists of only finitely many open sets (though each of those open sets may be infinite)).
A set $K \subset \C$ (or more generally, $K \subset \R^n$ for any $n \in \N$) is compact if and only if it is closed and bounded.
A set $K \subset \C$ is compact if and only if every sequence in $K$ has a subsequence converging to a limit in $K$.
If $K \subseteq \C$ is compact and $f : K \to \C$ is continuous, there are $z_1, z_2 \in K$ so that for all $z \in K$, $$\abs{f(z_1)} \leq \abs{f(z)} \leq \abs{f(z_2)}.$$ Relatedly, if $K \subseteq \C$ is compact and $f : K \to \C$, then $f(K)$ is also compact.
Connected
A non-empty set $S \subseteq \C$ is connected if whenever $U_1, U_2 \subseteq \C$ are disjoint open sets with $S \subseteq U_1 \cup U_2$, either $S \subseteq U_1$ or $S \subseteq U_2$.
A non-empty set $S \subseteq \C$ is path connected if whenever $z, w \in S$ there is a continuous function $f : [0, 1] \to S$ with $f(0) = w$ and $f(1) = z$ (a path in $S$ from $w$ to $z$).
If $U \subseteq \C$ is open, then $U$ is path connected if and only if it is connected.
If $S \subseteq \C$ is connected and $f : S \to \C$ is continuous, then $f(S)$ is also connected.