A Topological preliminaries

A.1 Metrics and related concepts

Definition A.1 Metric

A metric on a set $X$ is a function $\mathsf{d}\colon X \times X \to [0,\infty )$ such that for all $p_1, p_2, p_3 \in X$ we have

\begin{align*} \mathsf{d}(p_1, p_3) \le \; & \mathsf{d}(p_1,p_2) + \mathsf{d}(p_2,p_3) & \text{(triangle inequality)} \\ \mathsf{d}(p_1, p_2) = \; & \mathsf{d}(p_2,p_1) & \text{(symmetricity)} \\ \mathsf{d}(p_1, p_2) = \; & 0 \text{ if and only if } p_1 = p_2 . & \text{(separation of points)} \end{align*}

The set $X$ equipped with the metric $\mathsf{d}$ on it is called a metric space.

Lemma A.2 Metric in the complex plane

The formula

\begin{align*} \mathsf{d}(z,w) = |z - w| \qquad \text{ for } z,w \in \mathbb {C}\end{align*}

defines a metric on the complex plane $\mathbb {C}$.

(Thus $\mathbb {C}$ becomes a metric space. Also any subset of $\mathbb {C}$, in particular $\mathbb {R}\subset \mathbb {C}$, becomes a metric space when equipped with the metric given by the above formula restricted to the subset.)

Proof ▶

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Definition A.3 Ball (disk)

Let $X$ be a metric space with metric $\mathsf{d}\colon X \times X \to [0,\infty )$. Let $p_0 \in X$ be a point and let $r{\gt}0$.

The set

\begin{align*} \mathcal{B}(p_0; r) = \Big\{ p \in X \; \Big| \; \mathsf{d}(p,p_0) {\lt} r \Big\} \end{align*}

is called an open ball in $X$, centered at $p_0$, and with radius $r$.

The set

\begin{align*} \overline{\mathcal{B}}(p_0; r) = \Big\{ p \in X \; \Big| \; \mathsf{d}(p,p_0) \le r \Big\} \end{align*}

is called a closed ball in $X$, centered at $p_0$, and with radius $r$.

(In the case of the complex plane $\mathbb {C}$, the term disk is often used instead of the general metric space theory term ball.)

Definition A.4 Interior point

Let $X$ be a metric space, and $A \subset X$ a subset. A point $p \in A$ is said to be an interior point of $A$ if for some $r {\gt} 0$ we have $\mathcal{B}(p; r) \subset A$.

Definition A.5 Exterior point

Let $X$ be a metric space, and $A \subset X$ a subset. A point $p \in X \setminus A$ is said to be an exterior point of $A$ if for some $r {\gt} 0$ we have $\mathcal{B}(p; r) \subset X \setminus A$.

(It is easy to see that the exterior points of $A$ are exactly the interior points)

Definition A.6 Boundary

Let $X$ be a metric space, and $A \subset X$ a subset. A point $p \in X$ is said to be a boundary point of $A$ if for all $r {\gt} 0$ we have that $\mathcal{B}(p; r)$ contains points of $A$ and $X \setminus A$ (i.e. $\mathcal{B}(p; r) \cap A \ne \emptyset $ and $\mathcal{B}(p; r) \setminus A \ne \emptyset $).

The set of all boundary points of $A$ is denoted $\partial A$ and called the boundary of $A$.

(It is easy to see that the boundary $\partial A \subset X$ is exactly the set of points of $X$ which are neither interior nor exterior points of $A$.)

Definition A.7 Open set

Let $X$ be a metric space. A subset $U \subset X$ is said to be an open set if each point $p \in U$ is an interior point of $U$.

Definition A.8 Closed set

Let $X$ be a metric space. A subset $F \subset X$ is said to be a closed set if the complement $X \setminus F \subset X$ is an open set.

(Equivalently, each point $p \in X \setminus F$ in the complement of $F$ is an exterior point of $F$.)

Definition A.9 Boundedness

Let $X$ be a metric space. with metric $\mathsf{d}\colon X \times X \to [0,\infty )$.

A subset $A \subset X$ is bounded if there exists a number $M{\gt}0$ such that $\mathsf{d}(p,q) \le M$ for all $p,q \in A$. (If $X$ is nonempty, an equivalent definition would be that $A$ is bounded if it is a subset of some ball in $X$.)

A function $f \colon Z \to X$ with values in a metric space $X$ is bounded if the set $f[Z] \subset X$ of its values is a bounded subset of $X$.

(In the case $X = \mathbb {C}$ we have the following further characterizations: A subset $A \subset \mathbb {C}$ is bounded if and only if there exists an $R{\gt}0$ such that $|z| \le R$ for all $z \in A$. A function $f \colon Z \to \mathbb {C}$ is bounded if and only if there exists an $R{\gt}0$ such that $|f(z)| \le R$ for all $z \in Z$.)

A.2 Limits

Definition A.10 Limit

Let $X$ be a metric space and let $(x_n)_{n \in \mathbb {N}}$ be a sequence of points in $X$. We say that the sequence $(x_n)_{n \in \mathbb {N}}$ converges to a limit $x \in X$ if for any $\varepsilon {\gt} 0$ there exists an $N \in \mathbb {N}$ such that for all $n \ge N$ we have $x_n \in \mathcal{B}(x; \varepsilon )$ (i.e., $\mathsf{d}(x_n,x) {\lt} \varepsilon $). We then denote

\begin{align*} \lim _{n \to \infty } x_n = x . \end{align*}

(It is straightforward to check that the limit is unique if it exists.)

Let then $X$ and $Y$ be metric spaces, with respective metrics $\mathsf{d}_X$ and $\mathsf{d}_Y$, and let $f \colon X \to Y$ be a function. We say that the function $f$ has a limit $y \in Y$ at a point $p_0 \in X$ if for any $\varepsilon {\gt} 0$ there exists a $\delta {\gt} 0$ such that for all $p \in \mathcal{B}(p_0; \delta ) \setminus \left\{ p_0 \right\} $ we have $f(p) \in \mathcal{B}(y; \varepsilon )$. We then denote

\begin{align*} \lim _{p \to p_0} f(p) = y . \end{align*}

(It is straightforward to check that the limit is unique if it exists.)

(Equivalently, written in terms of distances, $\lim _{p \to p_0} f(p) = y$ means that for any $\varepsilon {\gt} 0$ there exists a $\delta {\gt} 0$ such that we have $\mathsf{d}_Y \big( f(p), y \big) {\lt} \varepsilon $ whenever $0 {\lt} \mathsf{d}_X (p,p_0) {\lt} \delta $.)

Lemma A.11 Limits in the complex plane

For a sequence $(z_n)_{n \in \mathbb {N}}$ of complex numbers we have

\begin{align*} \lim _{n \to \infty } z_n = z \end{align*}

if and only if

\begin{align*} \lim _{n \to \infty } \Re \mathfrak {e}(z_n) = \Re \mathfrak {e}(z) \quad \text{and} \quad \lim _{n \to \infty } \Im \mathfrak {m}(z_n) = \Im \mathfrak {m}(z) . \end{align*}

Let $X$ be a metric space, let $f \colon X \to \mathbb {C}$ a complex-valued function on $X$, and let $p_0 \in X$ be a point. Then we have

\begin{align*} \lim _{p \to p_0} f(p) = z \end{align*}

if and only if

\begin{align*} \lim _{p \to p_0} \Re \mathfrak {e}\big( f(p) \big) = \Re \mathfrak {e}(z) \quad \text{and} \quad \lim _{p \to p_0} \Im \mathfrak {m}\big( f(p) \big) = \Im \mathfrak {m}(z) . \end{align*}

Proof ▶

…

Lemma A.12 Operations with complex limits

Let $(z_n)_{n \in \mathbb {N}}$ and $(w_n)_{n \in \mathbb {N}}$ be complex number sequences converging to limits

\begin{align*} \lim _{n \to \infty } z_n = z \quad \text{and} \quad \lim _{n \to \infty } w_n = w . \end{align*}

Then we have

\begin{align*} \lim _{n \to \infty } (z_n + w_n) \, = \, z + w , \quad \lim _{n \to \infty } (z_n w_n) \, = \, z w , \quad \lim _{n \to \infty } \frac{z_n}{w_n} \, = \, \frac{z}{w} \; \text{ if $w \ne 0$}. \end{align*}

Let $X$ be a metric space, let $p_0 \in X$ be a point, and let $f,g \colon X \to \mathbb {C}$ be two complex-valued functions on $X$ such that

\begin{align*} \lim _{p \to p_0} f(p) = z \quad \text{and} \quad \lim _{p \to p_0} g(p) = w . \end{align*}

Then we have

\begin{align*} \lim _{p \to p_0} \big( f(p) + g(p) \big) \, = \, z + w , \quad \lim _{p \to p_0} \big( f(p) \, g(p) \big) \, = \, z w , \quad \lim _{p \to p_0} \frac{f(p)}{g(p)} \, = \, \frac{z}{w} \; \text{ if $w \ne 0$}. \end{align*}

Proof ▶

The arguments are similar to the proofs given in MS-C1541 Metric Spaces for the real-valued cases.

Definition A.13 Cauchy sequence

…

Lemma A.14 Every real Cauchy sequence converges

If a real number sequence $(x_n)_{n \in \mathbb {N}}$ is Cauchy, then it converges to a limit $\lim _{n \to \infty } x_n \in \mathbb {R}$.

(This property is known as completeness of the metric space $\mathbb {R}$.)

Proof ▶

See MS-C1541 Metric Spaces.

Lemma A.15 Every complex Cauchy sequence converges

If a complex number sequence $(z_n)_{n \in \mathbb {N}}$ is Cauchy, then it converges to a limit $\lim _{n \to \infty } z_n \in \mathbb {C}$.

(This property is known as completeness of the metric space $\mathbb {C}$.)

Proof ▶

See MS-C1541 Metric Spaces.

(Idea: This follows from Lemma A.14 by considering real and imaginary parts separately and picking a subsequence of a subsequence.)

A.3 Continuity

Definition A.16 Continuity

Let $X$ and $Y$ be metric spaces. A function $f \colon X \to Y$ is said to be continuous at a point $p_0 \in X$ if $\lim _{p \to p_0} f(p) = f(p_0)$.

(Equivalently, for every $\varepsilon {\gt} 0$ there exists a $\delta {\gt} 0$ such that for any $p \in \mathcal{B}(p_0; \delta )$ we have $f(p) \in \mathcal{B}(f(p_0); \varepsilon )$.)

A function $f \colon X \to Y$ is said to be continuous if it is continuous at every point $p_0 \in X$.

Lemma A.17 Continuity of complex-valued functions

Let $X$ be a metric space, and let $f \colon X \to \mathbb {C}$ be a complex-valued function on $X$. Then $f$ is continuous at $p_0 \in X$ if and only if its real and imaginary parts $p \mapsto \Re \mathfrak {e}\big( f(p) \big)$ and $p \mapsto \Im \mathfrak {m}\big( f(p) \big)$ are continuous at $p_0$.

Proof ▶

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Corollary A.18 Continuity of coordinate projections

The coordinate projections

\begin{align*} \Re \mathfrak {e}\colon \; & \mathbb {C}\to \mathbb {R}\qquad \qquad \text{ and } & \quad \Im \mathfrak {m}\colon \; & \mathbb {C}\to \mathbb {R}\\ & z \mapsto \Re \mathfrak {e}(z) & & z \mapsto \Im \mathfrak {m}(z) \end{align*}

are continuous functions.

Proof ▶

…

Lemma A.19 Operations with continuous complex-valued functions

Let $X$ be a metric space, let $p_0 \in X$ be a point, and let $f,g \colon X \to \mathbb {C}$ be two complex-valued functions on $X$ which are continuous at $p_0$. Then also the functions

\begin{align*} p \mapsto f(p) + g(p) \quad \text{and} \quad p \mapsto f(p) \, g(p) \end{align*}

are continuous at $p_0$.

If moreover $g(p_0) \ne 0$, then also the function $p \mapsto \frac{f(p)}{g(p)}$ is continuous at $p_0$.

Proof ▶

…

Lemma A.20 Continuity characterization

Let $X$ and $Y$ be metric spaces, and let $f \colon X \to Y$ be a function. Then the following are equivalent:

$f$ is a continuous function;
for every open set $V \subset Y$, the preimage $f^{-1}[V] = \big\{ x \in X \, \big| \, f(x) \in V \big\} $ is an open set in $X$;
for every closed set $A \subset Y$, the preimage $f^{-1}[A] = \big\{ x \in X \, \big| \, f(x) \in A \big\} $ is a closed set in $X$.

Proof ▶

See MS-C1541 Metric Spaces.

Lemma A.21 Composition of continuous functions

Let $X$, $Y$, and $Z$ be metric spaces, and let $f \colon X \to Y$ and $g \colon Y \to Z$ be functions. If $f$ is continuous at $x_0 \in X$ and $g$ is continuous at $f(x_0) \in Y$, then the composition $g \circ f \colon X \to Z$ is continuous at $x_0$.

(The composition $g \circ f$ is defined by the formula $(g \circ f)(x) = g \big( f(x) \big)$.)

Proof ▶

…

Corollary A.22 Real multivariate polynomials are continuous

Let $N \in \mathbb {N}$ be a natural number, and let $c_{n,m} \in \mathbb {R}$ be real numbers for $n,m \in \{ 0,1,\ldots ,N\} $. Then the function $p \colon \mathbb {C}\to \mathbb {R}$ defined by

\begin{align*} p(x + \mathfrak {i}y) = \sum _{m=0}^N \sum _{n=0}^N c_{m,n} \, x^m \, y^n \end{align*}

is continuous.

Proof ▶

See MS-C1541 Metric Spaces.

Definition A.23 Uniform continuity

Let $X$ and $Y$ be metric spaces. A function $f \colon X \to Y$ is uniformly continuous if for every $\varepsilon {\gt} 0$ there exists a $\delta {\gt} 0$ such that for any $p_0 \in X$ and $p \in \mathcal{B}(p_0; \delta )$ we have $f(p) \in \mathcal{B}(f(p_0); \varepsilon )$.

Lemma A.24 Uniform continuity implies continuity

If a function $f \colon X \to Y$ is uniformly continuous, then it is continuous.

Proof ▶

See MS-C1541 Metric Spaces.

(The easy proof is also a good exercise.)

A.4 Connectedness and path-connectedness

Definition A.25 Connectedness

A set $A \subset X$ in a metric space $X$ is disconnected if there exists a continuous surjective function $f \colon A \to \left\{ 0,1 \right\} $ onto the two-element discrete set $\left\{ 0,1 \right\} $. Otherwise $A$ is connected; then every continuous function $A \to \left\{ 0,1 \right\} $ must be either constant $0$ or constant $1$.

(The usual definition in topology textbooks reads slightly differently, but it is equivalent to the one we chose here by Lemma A.20.)

Definition A.26 Path-connectedness

A set $A \subset X$ in a metric space $X$ is path connected if for any two points $p, q \in X$ there exists a continuous function $\gamma \colon [0,1] \to X$ such that $\gamma (0) = p$ and $\gamma (1) = q$ (a parametrized path in $X$ starting from $p$ and ending at $q$).

Lemma A.27 Path-connectedness implies connectedness

If a metric space $X$ is path-connected, then it is connected.

Proof ▶

See MS-C1541 Metric Spaces.

Lemma A.28 Open connected sets are path-connected

Suppose that $U \subset \mathbb {C}$ is an open subset of the complex plane. Then $U$ is connected if and only if it is path-connected.

Proof ▶

See MS-C1541 Metric Spaces.

A.5 Compactness

Definition A.29 Compactness

Let $X$ be a metric space. A subset $K \subset X$ is compact if every sequence $(x_n)_{n \in \mathbb {N}}$ of points $x_n \in K$ has a subsequence $(x_{n_k})_{k \in \mathbb {N}}$ which converges to a limit $\lim _{k \to \infty } x_{n_k} \in K$ in the set $K$.

Theorem A.30 Bolzano-Weierstrass theorem

A subset $B \subset \mathbb {R}$ of the real line is compact if an only if it is closed and bounded.

A subset $A \subset \mathbb {C}$ of the complex plane is compact if an only if it is closed and bounded.

Proof ▶

See MS-C1541 Metric Spaces.

Theorem A.31 Boundedness of continuous functions on compacts

Suppose that $X$ is compact. Then every continuous function $f \colon X \to \mathbb {R}$ is bounded.

Proof ▶

…

Lemma A.32 On a compact domain continuity implies uniform continuity

If $X$ is compact and a function $f \colon X \to Y$ is continuous, then it is uniformly continuous.

Proof ▶

See MS-C1541 Metric Spaces.

Lemma A.33 Continuous bijection from a compact domain is a homeomorphism

Let $X$ and $Y$ be metric spaces and assume that $X$ is compact. Then for any continuous bijection $f \colon X \to Y$, also the inverse $f^{-1} \colon Y \to X$ is continuous.

Proof ▶

See MS-C1541 Metric Spaces.

Theorem A.34 Cantor’s intersection theorem [Palka1991, Thm II.4.5]

Let $X$ be a metric space. Suppose that $K_1, K_2, K_3, \ldots $ are nonempty compact subsets of $X$ nested so that $K_1 \supset K_2 \supset K_3 \supset \cdots $. Then the intersection $\bigcap _{n=1}^\infty K_n$ is nonempty.

Proof ▶

See MS-C1541 Metric Spaces.

A.6 Simple connectedness

Definition A.35 Path homotopy for closed paths

Let $X$ be a metric space and $\gamma _0 \colon [a,b] \to X$ and $\gamma _0 \colon [a,b] \to X$ two closed paths in $X$. If there exists a continuous function (called a homotopy)

\begin{align*} \Gamma \colon [0,1] \times [a,b] \to X \end{align*}

such that

\begin{align*} & \Gamma (0,t) = \gamma _0(t) \quad \text{ and } \quad \Gamma (1,t) = \gamma _1(t) \qquad \text{for all $t \in [a,b]$} \end{align*}

and

\begin{align*} \Gamma (s,a) = \Gamma (s,b) \qquad \text{for all $s \in [0,1]$,} \end{align*}

then we say that the closed paths $\gamma _0$ and $\gamma _1$ are homotopic.

Definition A.36 Contractible path

Let $X$ be a metric space. A closed path $\gamma \colon [a,b] \to X$ is called contractible if it is homotopic to a constant path.

Definition A.37 Simple connectedness

A metric space is said to be simply connected if every closed path $\gamma \colon [a,b] \to X$ in $X$ is contractible.