0Introduction

IB Metric and Topological Spaces

0 Introduction

The course Metric and Topological Spaces is divided into two parts. The first is

on metric spaces, and the second is on topological spaces (duh).

In Analysis, we studied real numbers a lot. We defined many properties such

as convergence of sequences and continuity of functions. For example, if (

x

n

) is

a sequence in R, x

n

→ x means

(∀ε > 0)(∃N)(∀n > N) |x

n

− x| < ε.

Similarly, a function f is continuous at x

0

if

(∀ε > 0)(∃δ > 0)(∀x) |x − x

0

| < δ ⇒ |f(x) − f(x

0

)| < ε.

However, the definition of convergence doesn’t really rely on

x

n

being real

numbers, except when calculating values of

|x

n

− x|

. But what does

|x

n

− x|

really mean? It is the distance between

x

n

and

x

. To define convergence, we

don’t really need notions like subtraction and absolute values. We simply need a

(sensible) notion of distance between points.

Given a set

X

, we can define a metric (“distance function”)

d

:

X × X → R

,

where

d

(

x, y

) is the distance between the points

x

and

y

. Then we say a sequence

(x

n

) in X converges to x if

(∀ε > 0)(∃N)(∀n > N) d(x

n

, x) < ε.

Similarly, a function f : X → X is continuous if

(∀ε > 0)(∃δ > 0)(∀x) d(x, x

0

) < δ ⇒ d(f(x), f(x

0

)) < ε.

Of course, we will need the metric

d

to satisfy certain conditions such as being

non-negative, and we will explore these technical details in the first part of the

course.

As people studied metric spaces, it soon became evident that metrics are not

really that useful a notion. Given a set

X

, it is possible to find many different

metrics that completely agree on which sequences converge and what functions

are continuous.

It turns out that it is not the metric that determines (most of) the properties

of the space. Instead, it is the open sets induced by the metric (intuitively, an

open set is a subset of

X

without “boundary”, like an open interval). Metrics

that induce the same open sets will have almost identical properties (apart from

the actual distance itself).

The idea of a topological space is to just keep the notion of open sets and

abandon metric spaces, and this turns out to be a really good idea. The second

part of the course is the study of these topological spaces and defining a lot of

interesting properties just in terms of open sets.