Symmetry of metric spaces.

Question

Does that just say that the distance from $a$ to $b$ equals the distance from $b$ to $a$? Is that the definition of symmetry in the metric spaces?

In case it is interesting for you, here is a discussion of spaces where the symmetry condition is omitted, only the remaining conditions from the definition of metric spaces are required: Examples of non symmetric distances. — Martin Sleziak, Sep 10 '15 at 11:52

score 1 · Answer 1 · answered Sep 10 '15 at 08:44

That may depend on where you got the term "symmetry of metric space" from. What you're describing is the symmetry axiom of metric spaces. The axioms for metric spaces are

$d(x,y) = 0 \Leftrightarrow x=y$ (coincidence axiom)
$d(x,y) = d(y,x)$ (axiom of symmetry)
$d(x,y) \le d(x,z) + d(z,x)$ (triangle axiom)

I've seen a reference that calls a space for "symmetric space" where only the two first axioms are required, but this seem to be non-standard. In that case every metric space will also be a so called "symmetric space".

Symmetry of metric spaces.

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