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.

Notice that this differs from the definition of metric space; since we can have $p(x,x)>0$.

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

Thank you.

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 $X$ generates a topology $τ_p$ on $X$ whose base is the family of open-balls $$B(x,\varepsilon)=\{y\in X:|p(x,y)-p(x,x)|<\varepsilon\}.$$

Thank you.

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 $X$ generates a topology $τ_p$ on $X$ whose base is the family of open-balls $$B(x,\varepsilon)=\{y\in X:|p(x,y)-p(x,x)|<\varepsilon\}.$$

Thank you.

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.

Notice that this differs from the definition of metric space; since we can have $p(x,x)>0$.

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

Thank you.

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 $τ_p$ on $X$ whose base is the family of open-balls . $$B(x,\varepsilon)=\{y\in X:|p(x,y)-p(x,x)|<\varepsilon\}$$.

Thank you  .

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 $X$ generates a topology $τ_p$ on $X$ whose base is the family of open-balls $$B(x,\varepsilon)=\{y\in X:|p(x,y)-p(x,x)|<\varepsilon\}.$$

Thank you.