=====Metric spaces=====

Abbreviation: **MetSp**

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

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

$d$ is \emph{symmetric}:  $d(x,y)=d(y,x)$

the \emph{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 \emph{...} 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====

[(Lastname19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]