Let {X,T} be a topology, T the set of open subsets of X.

Definition: Three points x, y, z of X are in *relation N* (Nxyz, read "x is nearer to y than to z") iff

- there is a basis
**B**of T and**b**in**B**such that x and y are in**b**but z is not and - there is no basis
**C**of T and**c**in**C**such that x and z are in**c**but not y.

For some topologies there are no points x, y, z in relation N, for example if T = {Ø,X} or T = P(X), but for others there are (e.g. for ones induced by a metric [my claim]).

Definition: A topology has *property M1* iff

(x)(y) ((z) (z ≠ x & z ≠ y) → Nxyz) → x = y

(This is an analogue of d(xy) = 0 → x = y, the best one I can imagine).

Definition: A topology has *property M2* iff

(x)(y)(z) Nxyz & Nyzx → Nzyx

(This is a kind of an analogue of d(xy) = d(yx), the best one I can imagine)

First (bunch of) question(s):

Properties

*M1*and*M2*do not capture the whole of the corresponding conditions of a metric. Can anyone figure out "better" definitions (e.g. an analogon of x = y → d(xy) = 0)?Can anyone figure out a

*property M3*that is an analogue of the triangle equality?

If it can be shown that no such property M3 is definable, the following becomes obsolete.

If such a definition can be made, we define:

Definition: A topology has *property M* (read "induces a metric") iff it has properties M1, M2, M3.

Second question:

Which topologies have property M, i.e. induce a metric? Are these "accidentally" exactly those that are induced by a metric?

x is closer to y than to zif....". – Theo Johnson-Freyd Dec 20 '09 at 3:16