Editing Metric space (section)

=== Definition ===
Formally, a '''metric space''' is an [[ordered pair]] {{math|(''M'', ''d'')}} where {{mvar|M}} is a set and {{mvar|d}} is a '''metric''' on {{mvar|M}}, i.e., a [[Function (mathematics)|function]]<math display="block">d\,\colon M \times M \to \mathbb{R}</math>satisfying the following axioms for all points <math>x,y,z \in M</math>:{{sfn|Burago|Burago|Ivanov|2001|p=1}}{{sfn|Gromov|2007|p=xv}}
# The distance from a point to itself is zero: <math display="block">d(x, x) = 0</math>
# (Positivity) The distance between two distinct points is always positive: <math display="block">\text{If }x \neq y\text{, then }d(x, y)>0</math>
# ([[Symmetric function|Symmetry]]) The distance from {{mvar|x}} to {{mvar|y}} is always the same as the distance from {{mvar|y}} to {{mvar|x}}: <math display="block">d(x, y) = d(y, x)</math>
# The [[triangle inequality]] holds: <math display="block">d(x, z) \leq d(x, y) + d(y, z)</math>This is a natural property of both physical and metaphorical notions of distance: you can arrive at {{mvar|z}} from {{mvar|x}} by taking a detour through {{mvar|y}}, but this will not make your journey any shorter than the direct path.

If the metric {{mvar|d}} is unambiguous, one often refers by [[abuse of notation]] to "the metric space {{mvar|M}}".

By taking all axioms except the second, one can show that distance is always non-negative:<math display="block">0 = d(x, x) \leq d(x, y) + d(y, x) = 2 d(x, y)</math>Therefore the second axiom can be weakened to <math display="inline">\text{If }x \neq y\text{, then }d(x, y) \neq 0</math> and combined with the first to make <math display="inline">d(x, y) = 0 \iff x=y</math>.<ref>{{Cite book |last=Gleason |first=Andrew |title=Fundamentals of Abstract Analysis |publisher=[[Taylor & Francis]] |year=1991 |edition=1st |pages=223 |doi=10.1201/9781315275444|isbn=9781315275444 |s2cid=62222843 }}</ref>