# continuity of b-metric

A b-metric is defined similar to a metric in which the triangle inequality is replaced by the inequality $$d(x,z)\leq s\Big[d(x,y)+d(x,z)\Big]\quad\forall\ x,y,z$$ where $s\geq1$.

There is an example of a b-metric which is not continuous.

My question: If the b-metric space is complete, can we conclude that the b-metric is continuous?

Any help would be appreciated. Thanks!

• Can you give the example or at least a reference? – Jochen Wengenroth Dec 8 '17 at 9:46
• Example 2.6 of  is an example of a b-metric which is not continuous.  S. K. Mohanta, Coincidence points and common fixed points for expansive type mappings in b-metric spaces, Iran. J. Math. Sci. Inform., Vol. 11, No. 1 (2016), pp. 101 – 113. DoI: 10.7508/ijmsi.2016.01-009. – mark haokip Dec 8 '17 at 11:25

Nope. Let $X$ be the set $\{0\} \cup \{\frac{1}{n}: n \in \mathbb{N}\} \subset \mathbb{R}$ together with one additional point $e$. The distance between two points neither of which is $e$ is just their usual distance in $\mathbb{R}$. Also set $d(e,e) = 0$, $d(e,0) = 2$, and $d(e,\frac{1}{n}) = 1$ for all $n$. This is a "b-metric" with $s = 2$ by inspection, and it is complete (any Cauchy sequence is either eventually constant or converges to $0$). But $\frac{1}{n} \to 0$ in the topology generated by $d$, while $d(e,\frac{1}{n}) \not\to d(e,0)$.
• Or just take $\mathbb R$ and define $d(x,y)=f(x,y)|x-y|$ with some wild (preferably not even existing without AC and CH) symmetric function $f:\mathbb R^2\to[1,s]$ with any $s>1$ you care about. I wonder how much more interesting questions get closed in no time and something like this survives... – fedja Dec 9 '17 at 4:22