Notes on Topology

David K. Zhang
Last Modified 2022-03-15

Introduction

Definition: Topology

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

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

Definition: Topological Space, Point, Open Set

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

Definition: Closed Set

Let (X,T)(X, T) be a topological space. A subset FXF \subseteq X is closed if XFTX \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(XA)=AX \setminus (X \setminus A) = A for all AXA \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)(X, T) be a topological space. A subset of XX is clopen if it is both open and closed.

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

Definition: Metric, Distance Function, Symmetry, Triangle Inequality

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

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

Definition: Metric Space, Underlying Set, Point

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

Definition: Open Ball

Let (X,d)(X, d) be a metric space, x0Xx_0 \in X, and rRr \in \R. The open ball of radius rr centered at x0x_0, denoted by Br(x0)B_r(x_0), is the subset of XX defined by

Br(x0){xX:d(x,x0)<r}. B_r(x_0) \coloneqq \{ x \in X : d(x, x_0) < r \}.