The Bonahon Metric and Topology

Abstract: In his book “Low-Dimensional Geometry: From Euclidean Spaces to Hyperbolic Knots”, Francis
Bonahon considers no structure more abstract than a metric space. He then needs to define a metric
on a quotient space, such as the torus obtained by identifying opposite sides of a rectangle. We explore
some quirky consequences of Bonahon’s definition of a (pseudo)-metric on a quotient space. We then
answer the question: does the topology defined by the Bohahon metric on a quotient space coincide
with the quotient topology?