There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different?
Thanks ahead.
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$\begingroup$There is some inconsistency in the literature, but most often:
A metric space is a set $S$ with a function $d:S\times S\to \mathbb R$ such that
$d(x,y)\ge 0$
$d(x,x)=0$
$d(x,y)=0\implies x=y$ (separation)
$d(x,y)=d(y,x)$ (symmetry)
$d(x,z)\le d(x,y)+d(y,z)$ (triangle inequality).
A quasi metric space satisfies the above except (possibly) for symmetry.
A pseudo metric space satisfies the above except (possibly) for separation.
A semi metric space often means a space satisfying the above except (possibly) for the triangle inequality. At times though, it also refers to a pseudo metric space, but sometimes it also refers to quasi metric space.
Generalized metric space or Lawvere space satisfies the above except (possibly) for separation and symmetry.
It is also possible to drop the $d(x,x)=0$ requirement, such things are called partial metric spaces and come in both pseudo/semi and quasi variants.
In relation to normed and semi-normed spaces, where the terminology is standard, defining $d(x,y)=\|x-y\|$ associated with a normed spaces a metric space and with a semi-normed space a metric space where $d(x,y)=0\implies x=y$ may fail. This could be seen as support for calling such spaces semi-metric spaces.
$\endgroup$ 5 $\begingroup$I think that
A semi-metric satisfies everything (in the definition of a metric) but not the triangle inequality.
A pseudo-metric satisfies everything except that it can have $d(x,y)=0$ for $x\ne y$. (It still must have $d(x,x)=0$.)
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