David K. Zhang

# Introduction

Definition: Topology

Let $X$ be a set. A topology on $X$ is a collection $T \subseteq \powerset{X}$ of subsets of $X$ that satisfies the following requirements:

• Empty set is open: $\varnothing \in T$.
• Whole set is open: $X \in T$.
• Closed under arbitrary unions: If $S \subseteq T$, then $\bigcup S \in T$.
• Closed under finite intersections: If $S \subseteq T$ and $\abs{S}$ is finite, then $\bigcap S \in T$.

Definition: Topological Space, Point, Open Set

A topological space is an ordered pair $(X, T)$ consisting of a set $X$, called the underlying set, and a topology $T$ on $X$. The elements of $X$ are called the points of the topological space $(X, T)$, and the elements of $T$ are called its open sets.

Definition: Closed Set

Let $(X, T)$ be a topological space. A subset $F \subseteq X$ is closed if $X \setminus F \in T$.

In other words, a set is closed if and only if its complement is open. Because taking complements is an involution (i.e., $X \setminus (X \setminus A) = A$ for all $A \subseteq X$), the converse is also true: a set is open if and only if its complement is closed.

Note that, in topology, “open” and “closed” are not opposites! It is possible for a set to be both open and closed, and it is also possible for a set to be neither. In fact, there is a special name for sets that are both open and closed.

Definition: Clopen Set

Let $(X, T)$ be a topological space. A subset of $X$ is clopen if it is both open and closed.

Every topological space $(X, T)$ has at least two clopen sets: the empty set $\varnothing$ and the set $X$ itself.

Definition: Metric, Distance Function, Symmetry, Triangle Inequality

Let $X$ be a set. A metric on $X$, also known as a distance function on $X$, is a function $d: X \times X \to \R$ that satisfies the following requirements:

• Positive-definiteness: For all $x, y \in X$, we have $d(x, y) \ge 0$, with $d(x, y) = 0$ if and only if $x = y$.
• Symmetry: $d(x, y) = d(y, x)$ for all $x, y \in X$.
• Triangle inequality: $d(x, z) \le d(x, y) + d(y, z)$ for all $x, y, z \in X$.

Definition: Metric Space, Underlying Set, Point

A metric space is an ordered pair $(X, d)$ consisting of a set $X$, called the underlying set of the metric space, and a metric $d$ on $X$. The elements of $X$ are called the points of the metric space.

Definition: Open Ball

Let $(X, d)$ be a metric space, $x_0 \in X$, and $r \in \R$. The open ball of radius $r$ centered at $x_0$, denoted by $B_r(x_0)$, is the subset of $X$ defined by

$B_r(x_0) \coloneqq \{ x \in X : d(x, x_0) < r \}.$