Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, for all positive rational $\alpha$: $(x,y)\in R_\alpha$ iff ${\mathrm{dist}}(x,y)<\alpha$. Is this the first paper where this algebraic system was defined?

Update 1: I need it because in my paper, I want to call these algebraic systems KSTT-systems. They satisfy axioms 0) $(x,x)\in R_\alpha $ for every $\alpha$, 1) $(x,y)\in R_\alpha$ iff $(y,x)\in R_\alpha$, 2) $(x,y)\in R_\alpha, (y,z)\in R_\beta\to (x,z)\in R_{\alpha+\beta}$, 3) $(x,y)\in R_\alpha\to (x,y)\in R_\beta$ for every $\beta\ge \alpha$. The original metric space $X$ can be elementary defined inside $A(X)$, and for every KSTT-system $A$, and an element $o\in A$, one can canonically (elementary) define a pointed metric space. This can be used to show that (modulo Continuum Hypothesis), for every metric space $X$ and every asymptotic cone $C$ of $X$, $C$ is isometric to any ultralimit of $C$. Thus, the only question remains: whether it is appropriate to call these KSTT-systems or somebody has introduced them earlier.

Update 2: I guess I was not clear enough. I need to know who was the first to consider metric spaces as algebraic systems with countable set of relations (as above). I am not interested in equivalent categories.

Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, for all positive rational $\alpha$: $(x,y)\in R_\alpha$ iff ${\mathrm{dist}}(x,y)<\alpha$. Is this the first paper where this algebraic system was defined?

Update 1: I need it because in my paper, I want to call these algebraic systems KSTT-systems. They satisfy axioms 0) $(x,x)\in R_\alpha $ for every $\alpha$, 1) $(x,y)\in R_\alpha$ iff $(y,x)\in R_\alpha$, 2) $(x,y)\in R_\alpha, (y,z)\in R_\beta\to (x,z)\in R_{\alpha+\beta}$, 3) $(x,y)\in R_\alpha\to (x,y)\in R_\beta$ for every $\beta\ge \alpha$. The original metric space $X$ can be elementary defined inside $A(X)$, and for every KSTT-system $A$, and an element $o\in A$, one can canonically (elementary) define a pointed metric space. This can be used to show that (modulo Continuum Hypothesis), for every metric space $X$ and every asymptotic cone $C$ of $X$, $C$ is isometric to any ultralimit of $C$. Thus, the only question remains: whether it is appropriate to call these KSTT-systems or somebody has introduced them earlier.

Update 2: I guess I was not clear enough. I need to know who was the first to consider metric spaces as algebraic systems with countable set of relations (as above). I am not interested in equivalent categories.

Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, for all positive rational $\alpha$: $(x,y)\in R_\alpha$ iff ${\mathrm{dist}}(x,y)<\alpha$. Is this the first paper where this algebraic system was defined?

Update: I need it because in my paper, I want to call these algebraic systems KSTT-systems. They satisfy axioms 0) $(x,x)\in R_\alpha $ for every $\alpha$, 1) $(x,y)\in R_\alpha$ iff $(y,x)\in R_\alpha$, 2) $(x,y)\in R_\alpha, (y,z)\in R_\beta\to (x,z)\in R_{\alpha+\beta}$, 3) $(x,y)\in R_\alpha\to (x,y)\in R_\beta$ for every $\beta\ge \alpha$. The original metric space $X$ can be elementary defined inside $A(X)$, and for every KSTT-system $A$, and an element $o\in A$, one can canonically (elementary) define a pointed metric space. This can be used to show that (modulo Continuum Hypothesis), for every metric space $X$ and every asymptotic cone $C$ of $X$, $C$ is isometric to any ultralimit of $C$. Thus, the only question remains: whether it is appropriate to call these KSTT-systems or somebody has introduced them earlier.

Let $(X, {\mathrm{dist}})$ be a metric space. In the paper by Kramer, Shelah, Tent and Thomas , they define an algebraic system $A(X)$ as the set $X$ with countably many binary relations $R_\alpha$, for all positive rational $\alpha$: $(x,y)\in R_\alpha$ iff ${\mathrm{dist}}(x,y)<\alpha$. Is this the first paper where this algebraic system was defined?

Update 1: I need it because in my paper, I want to call these algebraic systems KSTT-systems. They satisfy axioms 0) $(x,x)\in R_\alpha $ for every $\alpha$, 1) $(x,y)\in R_\alpha$ iff $(y,x)\in R_\alpha$, 2) $(x,y)\in R_\alpha, (y,z)\in R_\beta\to (x,z)\in R_{\alpha+\beta}$, 3) $(x,y)\in R_\alpha\to (x,y)\in R_\beta$ for every $\beta\ge \alpha$. The original metric space $X$ can be elementary defined inside $A(X)$, and for every KSTT-system $A$, and an element $o\in A$, one can canonically (elementary) define a pointed metric space. This can be used to show that (modulo Continuum Hypothesis), for every metric space $X$ and every asymptotic cone $C$ of $X$, $C$ is isometric to any ultralimit of $C$. Thus, the only question remains: whether it is appropriate to call these KSTT-systems or somebody has introduced them earlier.

Update 2: I guess I was not clear enough. I need to know who was the first to consider metric spaces as algebraic systems with countable set of relations (as above). I am not interested in equivalent categories.