## Metric spaces

Abbreviation: MetSp

### Definition

A metric space is a structure $\mathbf{X}=\langle X,d\rangle$, where $d:X\times X\to [0,infty)$ is a distance metric, i.e.,

points zero distance apart are identical: $d(x,y)=0\iff x=y$

$d$ is symmetric: $d(x,y)=d(y,x)$

the triangle inequality holds: $d(x,z)\le d(x,y)+d(y,z)$

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

Let $\mathbf{X}$ and $\mathbf{Y}$ be metric spaces. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:X\rightarrow Y$ that is continuous in the topology induced by the metric: $\forall z\in X\ \forall\epsilon>0\ \exists\delta>0\ \forall x\in X(0<d(x,z)<\delta\Longrightarrow d(h(x),h(z))<\epsilon$

### Definition

An is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Example 1:

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype higher-order

### Subclasses

[[Compact metric spaces]]

### Superclasses

[[Hausdorff spaces]] reduced type