Definition:
Let $X$ be an arbitrary set, $d:X\times X\to [0,\infty)$ be a mapping satisfying:
(a) $\forall_{x,y\in X}\; d(x,y)=0\iff x=y$;
(b) $\forall_{x,y\in X}\; d(x,y)=d(y,x)$;
(c) $\exists_{s\geq 1}\;\forall_{x,y,z\in X}\; d(x,y)\leq d(y,z)+sd(x,z)$.
Then $d$ is called a strong b-metric and $(X,d)$ is called a strong b-metric space.
Naturally, every metric space is a strong b-metric space as it fulfills (c) with $s=1$ (the classic triangle inequality).
Are there any natural examples of strong b-metric spaces which are not metric spaces? Note that every finite set $X$ equipped with a mapping $d$ fulfilling (a) and (b), fulfills (c) as well. Hence we are interested in examples on an infinite set $X$.
Best Answer
As recognized by Calvin Khor in the comments, one has $$d(x,y) \le \frac{s+1}{2}(d(x,z)+d(z,y)).$$ Thus, we have a quasi-metric. I am not sure, if this name is used in the case of metric spaces. However for norms this is a well-known concept, called quasinorm. For example, the spaces $L^p(\mu)$ and $l^p$ for $p \in (0,1)$ are quasi-normed spaces and even quasi-banachspaces. Much simpler one can take $d(x,y) = |x-y|^n$ on $\mathbb{R}$ to get a simple example.
However, there are no examples of b-metric spaces which are not metric spaces. The proof is not really complicated, see the paper On the Macias-Segovia Metrization of quasi-metric spaces by R. Aimar, B. Iaffei , L. Nitti, published in Revista de la Unión Matemática Argentina 41(2) (1998).