Necessity of the use of reals in the metric definition

general-topologymetric-spaces

A metric $d$ is a function $d:X \times X \to \mathbb{R}$ such that $d(x,y)\geq 0$ and equals $0$ iff $x=y$. $d(x,y)=d(y,x)$ and the triangle inequality holds. From these requirements, the only things that are used is that the codomain has a $0$, a $+$ operation and a linear order. So, it seems, we could in principle define a metric by a function $d: X \times X \to G$ such the same expressions hold, and where $G$ is an ordered group.

My question is, what usual theorems do we lose by picking that definition? In particular, some key questions come to mind

  1. If there is a metric on a space $X$, when allowing for other groups in the codomain, does that imply there is a metric with codomain $\mathbb{R}$? That is, does the collection of metrizable spaces expand with the new definition?
  2. A kind of converse to the previous one, for any infinite ordered group $G$, if there is metric with codomain $\mathbb R$, is there necessarily one with codomain $G$ that generates the same topology? (it being infinite is necessary as the trivial group satisfies all the metric properties but always generates the discrete topology).

Best Answer

The "ultimate generalisation" of such an idea (it's old) is due to Kopperman all topologies come from generalised metrics (Amer. Math. Monthly (95) 1988, nr 2, 89-97). I saw his talk on this around that time...

He considers a semigroup $A$ (so just an associative binary operation) with identity $0$ and $\infty \neq 0$ an absorbing element and calls it a value semigroup if

  • If $a+x=b$ and $b+y=a$, then $a=b$. In that case $a \le b$ iff $\exists x: a+x=b$ defines a partial order on $A$.
  • For each $b$ there is a unique $a$ so that $b+b =a$ (and we write $b = \frac12 a$).
  • For all $a,b$, $a \land b = \inf\{a,b\}$ exists.
  • For all $a,b,c$ we have $(a \land b) + c = (a+c) \land (b+c)$.

A set $P \subseteq A$, where $A$ is a value semigroup, is called a set of positives if

  • $a,b \in P \to a \land b \in P$.
  • $r \le a$ and $r \in P$ implies $a \in P$.
  • if $r \in P$ then $\frac12 r \in P$ as well.
  • if $a \le b+r$ for each $r \in P$, then $a \le b$.

Finally, if $X$ is a set, $A$ is a value semi-group, $P \subseteq A$ a set of positives, and $d: X \times X \to A$ a function that obeys $d(x,x)=0$ for all $x$ and $d(x,z) \le d(x,y) + d(y,z)$ for all $x,y,z \in X$, then $(X,A,P,d)$ is called a "continuity space".

For $x \in X, r \in P$ we define $B[x,r] = \{y \in X: d(x,y) \le r\}$ and then $\mathcal{T} = \{O \subset X\mid \forall x \in O: \exists r \in P: B[x,r]\subseteq O\}$ defines a topology on $X$ and (Kopperman's theorem) every topology on $X$ is of this form.

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