Metric spaces

Abbreviation: MetSp


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: $d(x,z)\le d(x,y)+d(y,z)$

Remark: This is a template. If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page.

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.


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: $\forall z\in X\ \forall\epsilon>0\ \exists\delta>0\ \forall x\in X(0<d(x,z)<\delta\Longrightarrow d(h(x),h(z))<\epsilon$


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

$...$ is …: $axiom$

$...$ is …: $axiom$


Example 1:

Basic results


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
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective


[[Compact metric spaces]]


[[Hausdorff spaces]] reduced type