Editing General topology (section)

==Metric spaces==
{{Main|Metric space}}
A '''metric space'''<ref>[[Maurice Fréchet]] introduced metric spaces in his work ''Sur quelques points du calcul fonctionnel'', Rendic. Circ. Mat. Palermo 22 (1906) 1–74.</ref> is an [[ordered pair]] <math>(M,d)</math> where <math>M</math> is a set and <math>d</math> is a [[metric (mathematics)|metric]] on <math>M</math>, i.e., a [[Function (mathematics)|function]]

:<math>d \colon M \times M \rightarrow \mathbb{R}</math>

such that for any <math>x, y, z \in M</math>, the following holds:

# <math>d(x,y) \ge 0</math> &nbsp;&nbsp;&nbsp; (''non-negative''), 
# <math>d(x,y) = 0\,</math> [[if and only if|iff]] <math>x = y\,</math> &nbsp;&nbsp;&nbsp; (''[[identity of indiscernibles]]''),
# <math>d(x,y) = d(y,x)\,</math> &nbsp;&nbsp;&nbsp; (''symmetry'') and
# <math>d(x,z) \le d(x,y) + d(y,z)</math> &nbsp;&nbsp;&nbsp; (''[[triangle inequality]]'') .

The function <math>d</math> is also called ''distance function'' or simply ''distance''. Often, <math>d</math> is omitted and one just writes <math>M</math> for a metric space if it is clear from the context what metric is used.

Every [[metric space]] is [[paracompact]] and [[Hausdorff space|Hausdorff]], and thus [[normal space|normal]].

The [[metrization theorems]] provide necessary and sufficient conditions for a topology to come from a metric.