A semimetric on a set \(X\) is a function \(d:X^2\to[0,\infty)\) that satisfies the first two conditions of a metric, \(x=y\) iff \(d(x,y)=0\) and \(d(x,y)=d(y,x)\), but not (necessarily) the triangle inequality. One topologizes a semimetric space in the natural way: \(U\) is open iff for every \(x\in U\) there is a positive \(r\) such that \(B(x,r)\subseteq U\). The possible absence of the triangle inequality makes it possible that the ‘open’ balls \(B(x,r)\) are no longer elements of the topology. After some remarks about the topology induced by the semimetric the authors concentrate on semimetric phenomena.
From [P. Tukia and J. Väisälä, Ann. Acad. Sci. Fenn., Ser. A I, Math. 5, 97–114 (1980; Zbl 0403.54005)] they adopt the definition of a quasisymmetric map between semimetric spaces. More precisely: given a homeomorphism \(\eta:[0,\infty)\to[0,\infty)\) a map \(f\) from \((X,d)\) to \((Y,\rho)\) is said to be \(\eta\)-quasisymmetric if for all \(t>0\) and all \(a\), \(b\), and \(x\) in \(X\) one has the implication if \(d(x,a)\le t\cdot d(x,b)\) then \(\rho\bigl(f(x),f(a)\bigr)\le\eta(t)\cdot\rho\bigl(f(x),f(b)\bigr)\). The remainder of the paper is devoted to determining when such maps preserve structural properties of the domain, such as having a triangle function (a function \(\Phi\) that satisfies \(d(x,z)\le\Phi\bigl(d(x,y),d(y,z)\bigr)\) for all \(x\), \(y\), and \(z\)), satisfying Ptolemy’s inequality (\(d(x,z)\cdot d(t,y)\le d(x,y)\cdot d(t,z) + d(x,t)\cdot d(y,z)\) for all \(x\), \(y\), \(z\), and \(t\)), or betweenness relations. The final sections are devoted to a generalization of Theorem 2.5 from [loc. cit.] and to weak similarities between semimetric spaces.