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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]]
		+	)]
		+
		+