This is a revised version of a question I already posted, but which patently was ill posed. Please give me another try.

For comparison's sake, the axioms of a metric:

Axiom A1: $(\forall x)\ d(x,x) = 0$

Axiom A2: $(\forall x,y)\ d(x,y) = 0 \rightarrow x = y$

Axiom A3: $(\forall x,y)\ d(x,y) = d(y,x)$

Axiom A4: $(\forall x,y,z)\ d(x,y) + d(y,z) \geq d(x,z)$

Let $T$ = {X,T} be a topology, $B$ a base of $T$, $x, y, z$ $\in$ X

Definition D0: $x$ **is nearer to** $y$ **than to** $z$ **with respect to** $B$ ($N_Bxyz$) iff $(\exists b \in B)\ x, y \in b \ \& \ z \notin b \ \&\ (\nexists b \in B)\ x, z \in b \ \& \ y \notin b$

Definition D1: $B$ is **pre-metric _{1}** iff $(\forall x,y)\ x \neq y \rightarrow N_Bxxy$

Definition D2: $B$ is **pre-metric _{2}** iff $(\forall x,y,z)\ ((z \neq x\ \&\ z \neq y) \rightarrow N_Bxyz) \rightarrow x = y$

Definition D3: $B$ is **pre-metric _{3}** iff $(\forall x,y,z)\ N_Bzyx \rightarrow (N_Byxz \rightarrow N_Bxyz)$

Definition: $T$ is pre-metric_{i} iff $(\exists B)\ B$ is pre-metric_{i} (i = 1,2,3).

Definition: $B$ is pre-metric iff $B$ is pre-metric_{1}, pre-metric_{2} and pre-metric_{3}.

Definition: $T$ is pre-metric iff $(\exists B)\ B$ is pre-metric.

Remark: D1 is an analogue of axiom A1, D2 of axiom A2, D3 of axiom A3.

Remark: $T$ is pre-metric_{1} iff $T$ is T_{1} *[not quite sure]*.

Remark: If $T$ is induced by a metric, then $T$ is pre-metric.

**Question: Can a property pre-metric_{4} be defined such that $T$ induces a metric iff $T$ is induced by a metric** with

Definition: $B$ is **metric** iff $B$ is pre-metric and pre-metric_{4}.

Definition: $T$ **induces a metric** iff $(\exists B)\ B$ is metric.

Remark: Property pre-metric_{4} should be an analogue of A4 (the triangle inequality).

If provably no such property can be defined does this shed a light on the difference (an asymmetry) between topologies and metric spaces? ("It's the triangle inequality, that cannot be captured topologically.")