Editing Metric space (section)

==Constructions==
===Product metric spaces===
{{main|Product metric}}
If <math>(M_1,d_1),\ldots,(M_n,d_n)</math> are metric spaces, and {{mvar|N}} is the [[Euclidean norm]] on <math>\mathbb R^n</math>, then <math>\bigl(M_1 \times \cdots \times M_n, d_\times\bigr)</math> is a metric space, where the [[product metric]] is defined by
<math display="block">d_\times\bigl((x_1,\ldots,x_n),(y_1,\ldots,y_n)\bigr) = N\bigl(d_1(x_1,y_1),\ldots,d_n(x_n,y_n)\bigr),</math>
and the induced topology agrees with the [[product topology]].  By the equivalence of norms in finite dimensions, a topologically equivalent metric is obtained if {{mvar|N}} is the [[taxicab norm]], a [[Norm (mathematics)#p-norm|p-norm]], the [[maximum norm]], or any other norm which is non-decreasing as the coordinates of a positive {{mvar|n}}-tuple increase (yielding the triangle inequality).

Similarly, a metric on the topological product of countably many metric spaces can be obtained using the metric
<math display="block">d(x,y)=\sum_{i=1}^\infty \frac1{2^i}\frac{d_i(x_i,y_i)}{1+d_i(x_i,y_i)}.</math>

The topological product of uncountably many metric spaces need not be metrizable. For example, an uncountable product of copies of <math>\mathbb{R}</math> is not [[first-countable space|first-countable]] and thus is not metrizable.

===Quotient metric spaces===
If {{mvar|M}} is a metric space with metric {{mvar|d}}, and <math>\sim</math> is an [[equivalence relation]] on {{mvar|M}}, then we can endow the quotient set <math>M/\!\sim</math> with a pseudometric. The distance between two equivalence classes <math>[x]</math> and <math>[y]</math> is defined as
<math display="block">d'([x],[y]) = \inf\{d(p_1,q_1)+d(p_2,q_2)+\dotsb+d(p_{n},q_{n})\},</math>
where the [[infimum]] is taken over all finite sequences <math>(p_1, p_2, \dots, p_n)</math> and <math>(q_1, q_2, \dots, q_n)</math> with <math>p_1 \sim x</math>, <math>q_n \sim y</math>, <math>q_i \sim p_{i+1}, i=1,2,\dots, n-1</math>.{{sfn|Burago|Burago|Ivanov|2001|loc=Definition 3.1.12}} In general this will only define a [[pseudometric space|pseudometric]], i.e. <math>d'([x],[y])=0</math> does not necessarily imply that <math>[x]=[y]</math>. However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces), <math>d'</math> is a metric.

The quotient metric <math>d'</math> is characterized by the following [[universal property]]. If <math>f\,\colon(M,d)\to(X,\delta)</math> is a metric (i.e. 1-Lipschitz) map between metric spaces satisfying {{math|''f''(''x'') {{=}} ''f''(''y'')}} whenever <math>x \sim y</math>, then the induced function <math>\overline{f}\,\colon {M/\sim}\to X</math>, given by <math>\overline{f}([x])=f(x)</math>, is a metric map <math>\overline{f}\,\colon (M/\sim,d')\to (X,\delta).</math>

The quotient metric does not always induce the [[quotient topology]].  For example, the topological quotient of the metric space <math>\N \times [0,1]</math> identifying all points of the form <math>(n, 0)</math> is not metrizable since it is not [[first-countable space|first-countable]], but the quotient metric is a well-defined metric on the same set which induces a [[comparison of topologies|coarser topology]].  Moreover, different metrics on the original topological space (a disjoint union of countably many intervals) lead to different topologies on the quotient.<ref>See {{harvnb|Burago|Burago|Ivanov|2001|loc=Example 3.1.17}}, although in this book the quotient <math>\N \times [0,1]/\N \times \{0\}</math> is incorrectly claimed to be homeomorphic to the topological quotient.</ref>

A topological space is [[sequential space|sequential]] if and only if it is a (topological) quotient of a metric space.<ref>Goreham, Anthony. [http://at.yorku.ca/p/a/a/o/51.pdf Sequential convergence in Topological Spaces] {{webarchive|url=https://web.archive.org/web/20110604232111/http://at.yorku.ca/p/a/a/o/51.pdf |date=2011-06-04 }}. Honours' Dissertation, Queen's College, Oxford (April, 2001), p. 14</ref>