# Differences

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+ | =====Metric spaces===== | ||

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+ | 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., | ||

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

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

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+ | 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. | ||

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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. | ||

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+ | ==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: | ||

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+ | ====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]] | | | ||

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+ | ====Subclasses==== | ||

+ | [[Compact metric spaces]] | ||

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+ | |||

+ | ====Superclasses==== | ||

+ | [[Hausdorff spaces]] reduced type | ||

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+ | ====References==== | ||

+ | |||

+ | [(Lastname19xx> | ||

+ | F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] | ||

+ | )] | ||

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