Let $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)\enspace p(x,y)=p(y,x)\hspace{1,2cm}\\ \hspace{0,2cm}&3)\enspace p(x,z)\leq p(x,y)+p(y,z) \end{align}Then the pair $(X,p)$ is said to be a metric-like space.

I want to show please that each metric-like $p$ on $X$ gererates a topology $τ_d$ on $X$ whose base is the family of open-balls . $$B(x,\varepsilon)=\{y\in X:|d(x,y)-d(x,x)|<\varepsilon\}$$.

Thank you .