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: $d(x,z)\le d(x,y)+d(y,z)$
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.
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
Trace: » metric_spaces