Real metric and pseudometric convention

Let $X$ be a set and use the ordered real field of The real numbers, Complete ordered field (least-upper-bound property) and Ordered field. In the convention of Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, a real metric is a function $d:X\times X\to \mathbb{R}$ satisfying, for all $x,y,z\in X$, separation $d(x,y)=0⟺x=y$, symmetry $d(x,y)=d(y,x)$ and the triangle inequality $d(x,z)\le d(x,y)+d(y,z)$. A real pseudometric replaces separation by $d(x,x)=0$ and retains the other two axioms. All values are finite real numbers. Neither definition assumes nonnegativity as an additional axiom.

For either structure, put $B_d(x,r)=\{y\in X:d(x,y)<r\}$ for real $r>0$. The induced topology consists of the subsets $O\subseteq X$ for which every $x\in O$ admits $r>0$ with $B_d(x,r)\subseteq O$. After the nonnegativity deduction in the well-definedness lemma, these balls form a basis: $x\in B_d(x,r)$ by reflexivity, and if $z\in B_d(x,r)\cap B_d(y,s)$, the positive radius $t=\min \{r-d(x,z),s-d(y,z)\}$ has $B_d(z,t)$ contained in that intersection by the triangle inequality. This argument also works for pseudometrics and for empty $X$ (the empty topology). No infinite-distance, semimetric or quasimetric convention is consumed.