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

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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 …: $axiom$

Examples

Example 1:

Basic results

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
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Subclasses

[[Compact metric spaces]]

Superclasses

[[Hausdorff spaces]] reduced type

References