Topology/Lesson 3

Metric Space

A metric space has very many useful properties and they are very good for beginning topology students to study because of their relative intuitive nature.

Definition

Let ${\displaystyle X}$ be a set. Then a metric on the set ${\displaystyle X}$ is a function ${\displaystyle d:X\times X\to \mathbb {R} }$ such that

1. ${\displaystyle d(x,y)\geq 0}$ and equality holds if and only if ${\displaystyle x=y.}$
2. ${\displaystyle d(x,y)=d(y,x)}$ (i.e., ${\displaystyle d}$ is symmetric).
3. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z)}$ (the triangle inequality).

We can define a topology on the set ${\displaystyle X}$ using the metric ${\displaystyle d}$ as follows. For ${\displaystyle x\in X}$ and ${\displaystyle r>0}$ define ${\displaystyle B(x,r)=\{y\in X\mid d(x,y) Then take the collection ${\displaystyle \{B(x,r)\mid x\in X,\ r>0\}}$ as a basis. Call this topology ${\displaystyle {\mathcal {T}}_{d}.}$

If ${\displaystyle (X,{\mathcal {T}})}$ is a topological space, then we say that it is metrizable if there is a metric ${\displaystyle d:X\times X\to \mathbb {R} }$ such that ${\displaystyle {\mathcal {T}}={\mathcal {T}}_{d}.}$ If ${\displaystyle X}$ is metrizable and ${\displaystyle d}$ is such a metric then we call the pair ${\displaystyle (X,d)}$ a metric space.

Examples

1. The most obvious example is the space ${\displaystyle \mathbb {R} }$ together with the metric ${\displaystyle d(x,y)=|x-y|}$ which is called the standard metric on ${\displaystyle \mathbb {R} .}$
2. Any set has a metric on it. As an example, let ${\displaystyle X}$ be any set and let ${\displaystyle d}$ be given by ${\displaystyle d(x,y)=\left\{{\begin{array}{ll}1&x\neq y\\0&x=y\end{array}}\right..}$ It is left as an exercise to check that this is a metric, called the discrete metric because the topology it gives ${\displaystyle X}$ is the discrete topology.

Exercises

1. Show that the discrete metric is a metric and that it gives the discrete topology on ${\displaystyle X.}$

Isometries

The "isomorphism" in the category of topological spaces is homeomorphism, as we saw in Lesson 2. The "isomorphism" in the category of metric spaces is called "global isometry" and is much more strict than homeomorphism.

Definition

Let ${\displaystyle (X,d)}$ and ${\displaystyle (Y,\rho )}$ be metric spaces. Then a function ${\displaystyle f:X\to Y}$ is called an isometry if for all ${\displaystyle x,y\in X}$ we have ${\displaystyle \rho (f(x),f(y))=d(x,y)}$ (i.e. if distances are preserved). If ${\displaystyle f}$ is also surjective, then we say that ${\displaystyle f}$ is a global isometry.

Examples

1. If ${\displaystyle (X,d)}$ is any metric space, then the identity map is an isometry.
2. If ${\displaystyle (Y,d)}$ is a metric space and ${\displaystyle X\subset Y}$ then the inclusion map ${\displaystyle i:X\to Y}$ is an isometry.

Exercises

1. Show that every isometry is injective and therefore every global isometry is bijective.
2. Show that every isometry is an embedding and therefore every global isometry is a homeomorphism.
3. Find a function that is a homeomorphism but not an isometry.