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metric_spaces [2010/07/29 15:46] (current)
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 +=====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: $d(x,z)\le d(x,y)+d(y,z)$
 +
 +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 \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]]
 +)]
 +
 +