# Notes on Topology

Last Modified 2022-03-15

# Introduction

**Definition: Topology**

Let * topology* on

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

**Definition: Topological Space, Point, Open Set**

A * topological space* is an ordered pair

*, and a topology*

**underlying set***of the topological space*

**points***.*

**open sets****Definition: Closed Set**

Let * closed* if

In other words, a set is closed if and only if its complement is open. Because taking complements is an involution (i.e.,

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 * clopen* if it is both open and closed.

Every topological space

**Definition: Metric, Distance Function, Symmetry, Triangle Inequality**

Let * metric* on

*on*

**distance function**: For all**Positive-definiteness**$x, y \in X$ , 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

*of the metric space, and a metric*

**underlying set***of the metric space.*

**points****Definition: Open Ball**

Let * open ball* of radius