Question

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

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

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

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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]).

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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).

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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)

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First (bunch of) question(s):

1. 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)?

2. 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:

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

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Second question:

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

Answer 1

Your condition 1 is satisfied for all triples $x,y,z\in X$ such that $z\not\in\{x,y\}$ if the space is $T_1$.

Maybe reading a bit about uniform spaces and the corresponding metrizability results will be of help.

Answer 2

I'm not sure your "nearness" relation is the right definition to make. It makes sense on the face of it, but for example I think you can generate the usual topology on the plane by tubular neighborhoods of half-circles. For three points x,y,z on a line in that order, I don't think you get any relationship. Using the usual disk basis you'd satisfy condition (1) for Nxyz, but you could violate condition (2) using the other basis I just suggested, if the tubular neighborhood is sufficiently small.