It is a very important question, since it tries to relate the concept of a linear space (dimension) with the core of Geometry. Probably we have learnt to think by the Euclidean way, in its generalized form (Riemannian), which demands a linear space attribute. Personally I know only some strange topological spaces as alternative, but also there we find the concept of linearity. I have similar questions:

https://www.researchgate.net/post/What_is_the_essential_difference_between_Algebra_and_Topology?

https://www.researchgate.net/post/Are_you_satisfied_with_our_linear_science?

I suppose that we cannot think without a linear way...

I agree, that linearity is important property. However, the most important property of any conception is its monism. It means, that the conception contains only one fundamental quantity. All other quantities of the conception are derivative, i.e. they are expressed via the fundamental quantity. For instance, geometry, as a science on shape of geometrical objects and their mutual disposition, can be presented in the monistic form. It is the metrical approach to geometry, when all definitions and geometrical quantities ( including dimension ) are expressed via distance between any two points of the point set, where the geometry is given. The proper Euclidean geometry GE  can be presented as a monistic conception with the fundamental quantity: world function  \sigma = \rho 2 /2,  where \rho is the dstance between two points.

Monism is important,  because it admits one to generalize GE replacing the world function of GE by  world function of a generalized geometry G . If the geometry GE  is pluralistic, i.e. there are several fundamental independent quantities, then generalizing GE ,  one needs to generalize all fundamental quantities. This generaizaation is to be concerted, in order that the generalized geometry G be consistent. To obtain a consistent generalized geometry G is very difficult task. For instance the Riemannian geometry is inconsistent. Consistency has been proved only for proper Euclidean geometry GE.

All operation of the linear vector space (scalar product and linear dependence) can be expessed in terms of the world function and only in terms of the world function. In particular, dimension of the proper Euclidean geometry can be expessed in terms of the Euclidean world function. However, at this expression the specific properties of the Euclidean world function are used. If the world function of the generalized geometry G has not these specific properties, then the generalized geometry has not any definite dimension, but it is a valuable geometry.

To my opinion this question is not well-posed, since the word "geometry" has no exact mathematical meaning and cannot have it in principle. Various geometries one may meet in contemporary mathematics are by its nature, so to say, road maps for various kind of mathematical reasonings. For instance, if a metric space is a geometrical object, then the answer to the question is banally positive. On the other hand, Riemannian geometry was developped as an instrument of metric mesurements on manifolds. So, the notion of a manifold and hence of its dimension is unevodable in this context. By the way, no coordinates on the manifold are required.

 Poisson (symplectic) geometry is  the geometry of classical dynamics. This geometry can be developped by starting only from the algebra of observables. In this approach  manifolds appear to be the spectra of  algebras of observables. In a similar manner one can interpret the kinetic energy as a Riemannian metric on the configuration space of a classical mechanical system, which, in its turn, is the spectrum of an algebra of observables. Very roughly speaking, in such a context the dimension is simply the number of instruments in the laboratory. In this situation the asked question sounds as : whether classical phisics be developped by physisits who cannot count the number of mesurement instruments in their laboratory.

This approach can be developped as well in field theory, but in this case it requires the new language of Secondary Calculus.

When we talk about manifold, it is a simplifying assumption for it to be n dimensional, where n is a constant.  In differential geometry, when one finds the zero dynamics of a nonlinear system, it may be defined on a manifold where the dimension at each point in the manifold may be different.  It is not essential for the dimension at each point in the manifold to be finite either.  One may consider manifolds where locally it is homeomorphic to a Hilbert space or Banach space.  This hasn't been done since differential geometry is still under development.  We need to work out the properties of an n dimensional manifold first before looking at infinite dimensional manifolds.  

Vinogradov expressed the viewpoint of mathematician, who systematize sciences on methods of investigations, whereas I (as a physicist) systematize sciences over subjects, which is investigated by the science.  The primary meaning of term "geometry" was a science on shape of geometrical objects and on mutual disposition of them. After two thousand years of the Euclidean geometry investigation, mathematicians decided, that the geometry is a kind of logical construction, and any generalized geometry is always a logical construction. If a science, investigating properties of the real space or space-time is not a logical construction, than it is not a geometry and it is not a mathematics at all. On the conrrary, if a science useses a logical construction, which is close to Euclidean geometry, it is qualified as a geometry, even if  this construction does not investigate shape of geometrical objects and their mutual disposition (for instance, symplectic geometry)

According to viewpoint of mathematicians a science, where there are no theorems and deductions, cannot be a geometry, because in this science the matematicians have no subject for investigation. Activity of mathematicians is a proof  of different theorems, and it is no importance, whether these theorems any real situation. On the contrary, physicist, investigating propertie of real space-time, needs an instrument, which admits him investigate  the space-time properlies. It is of no importance, whether or not this instrument forms a logical construction.

Unfortunately, the different understanding of the term "geometry" is not only a question of terminology. Let us consider a simple example of a discrete space- time geometry. The space-time geometry is discrete, if for any points P,Q the quantity |\rho(p,Q)| does not belong to interval (0,\lambda), where \rho (P,Q) is the distance between the points P,Q and \lambda is a minimal length. Mathematicians and physicists solve differently the problem of the discrete geometry construction. Considering the metric \rho as Euclidean (or Minkowskian) metric, mathematicians impose restrictions on the point set, where the geometry is given. As a result they obtain so called " geometry on lattice", which is not uniform and isotropic.

Physicist solve the problem with the discrete geometry as follows. Fixing the pointset in the same way, as it take place in the geometry of Minkowski, the physicist looks for the form of the world function \sigma=\rho2/2. The result has the form 

\sigma = \sigmaM+\lambda2 \2 sgn(\sigmaM)

where \sigmaM is the world function of the Minkowskian geometry. Such a discrete geometry is uniform and isotropic. Besides, connecting \lambda with the quantum constant \hbar, one can explain quantum effects as geometrical effectos of the discrete space-time geometry.

As to terminology, I use terms "mathematical geometry" and "physical geometry" for two different approaches to geometry. See the paper " What is geometry ? Physical geometry and mathematical geometry as two aspects of the proper Euclidean geometry"  http://gasdyn-ipm.ipmnet.ru/~rylov/tg1e.pdf

I read that paper:

http://arxiv.org/pdf/math/0511575v1.pdf

but i didn't understand the reason for introducing a similar to metric gij function like the world function.

It is possible to use metric. But world function is simple from technical viewpoint. Besides, the world function is always real, whereas the metric is imaginary  for spacelike distances in the Minkowskian (or Riemannian) geometry. The world function was introduced by J.L.Singe in 1931 for description of Riemannian geometry. 

John Wheeler's old "pregeometry" research programme might be of interest:

   http://en.wikipedia.org/wiki/Pregeometry_(physics)

http://en.wikipedia.org/wiki/Pregeometry_%28physics%29

I do not think that the "pregeometry" is simpler, than geometry. I do not like to explain a simple thing via more complicated one.

This discussion may perhaps have some relevance to your question:

http://math.stackexchange.com/questions/56986/differential-forms-on-fuzzy-manifolds

Dimension is a topological property (this is a non trivial fact[1]). The dimension of a Riemannian manifold does not come from the metric structure, but you start out with a topological space, build up from local sets that are homeomorphic with open balls in R^n (i.e. is a topological manifold) . The statement is that the n does not depend on the way you build up the space but is an intrinsic property of its topological structure.  Then if you have chosen transition functions that are differentiable, the manifold is not just a topological manifold, but a differentiable one and you have a nice tangent bundle of rank n, i.e. each tangent space has linear dimension n but is the same n.. Then if you chose a Riemannian metric, you get a Riemannian manifold. You can now define the dimension by the fact that the volume of a ball of radius r growth as r^n, but again it is the same n. The Riemannian metric also makes the manifold into a metric space but that metric is compatible with the topology which you started with so it is not the metric which gives the manifold its dimension n, it is just compatible with that and so gives the same topological dimension n.   

[1] To see that the dimension is topological property, you need a bit of algebraic topology. E..g. (unless M is zero dimensional which is easy to define topologically)  n is defined by the property that for every x \in M,  H_i(M, M-{x}) = 0 for i != n, and Z  for i = n. , where H_i is the i'th homology group.  

Dear Rogier, Any generalized geometry arises as a result of a generalization of the proper Euclidean geometry GE.  The generalization of GE depends on the GE presentation. If the presentation is pluralistic, i.e. it contains many fundamental independent quantities and concepts, then at the modification one  needs to modify any fundamental properties independently. The modification must be produced in such a way, that the modified geometry G be consistent. It is very difficult problem. Even the Riemannian geometry is inconsistent. The only possible way of generalization is founded on the monistic presentation of GE, when only one fundamental quantity lies in the foundation of GE. Such a presentation of GE contains the Euclidean metric as a basic quantity of GE. All other geometric quantities (including dimension) are expressed via the Euclidean metric \rhoE. 

Modification of GEconsists in replacement of the Euclidean metric \rhoE by the metric \rho of  G in all expressions of geometric quantities via \rhoE.  Such a replacement changes properties of geometrical quantities. Geometrical quantities of GE are different . Some of them do not depend on the form the world function \sigma=\rho2/2. For instance, the scalar product  (AB.PQ) of two vectors AB and PQ.

It has the form 

(AB.PQ) = \sigma (A,Q) +\sigma (B,P) - \sigma (A,P) - \sigma (B,Q)

for any two vectors AB and PQ in any generalized geometry.

However,  there are such geometric quantities in GE, which depends on the form of metric \rhoE. Such a quantity cannot be introduced in the generalized geometry G.  As a result the geometry G exists without such quantities, although G is a valueable geometry. Dimension n is such a quantitity, which can be introduced only in some generalized geometries. For instance, the dimension n cannot be introduced in the discrete space-time geometry, which is described by the world function \sigma = \sigmaM +\lambda2sgn(\sigmaM), where \sigmaM is the world function of the Minkowskian geometry, and  \lambda  is elementary length of the discrete space-time geometry. As a result the dimension cannot be introduced exactly in the discrete geometry. It can be introduced only approximately.

In the modified geometry G  the relation of equivalence is intraansitive, generally speaking, and the geometry G is nonaxiomatizable. But it is of now importance, because the concept of axiomatizability  relates to the method of the geometry constraction, but not to the geometry itself.

It depends on what is meant by "geometry". In graph theory we consider a set of "points" and a set of point pairs, which we may call "edges". If we are willing to call a graph a "geometry" then, in general, "dimensionality" of the graph is not a well-defined concept (though particular cases of finite geometries of a given dimensionality can be represented by a graph). One can ask what is minimal dimension dimensionality of a continuous topological space in which a particular graph can be embedded without intersections of edges. But this question goes beyond  the primitive "geometry" of the graph itself.

In mathematical point of you the answer for your question is yes.  Dimension, geometry, manifold etc. are only words. In mathematics, the word "dimension" has at least  three independent usage:

1. It means a concept which is defined in an axiomatic structure (in topology, in linear algebra, in measure theory, in a finite structures e.g. in a matroids ...).

2. The axioms of a structure contains such informations from which we can think to an apriori concept of dimension. (Axiomatic building up of the "3-dimensional" Euclidean geometry.)

3. In the definition of the structure we use a concept of dimension defined in an other structure. (topological manifold of dimension n, Remannian geometry of dimension n, n-dimensional Euclidean space etc..)

In 1. the concept of dimension is a tool, only. In 2. it is an inner property of the structure which we can call "dimension" however we can define another number as "dimension" if we want. In 3. on the borrowed concept of "dimension" determines the properties of the structure and it is not deduced for the other properties of the structure.  

Dear Eric,  Unfortunately mathematicians do not able to construct a generalized gtometry as a deformation of the proper Euclidean geometry. The can only sew a generalized geometry from pieces of the Euclidean geometry. For such a sewing they needs a topology. If a generalized geometry is constructed as a deformation of the Euclidean geometry, it obtains new properties. It may be multivariant. It may have no definite dimension, which is defined in Euclidean geometry  as maximal number of linear independent vectors. 

For details, see, for instance  "Geometry without topology" http://arXiv.org/abs/math.MG/0002161

I read your paper in arxiv and I found a lot of logical holes. The same in your commentaires. 

Definition 1.1 and 1.2 are ok. They are the classical definitions of metric space and subspace of a metric space. In 1.3 you said that P is a n+1-order metric subspace if P has n elements. 

Definition 1.4. what did you mean with the expression "to have some metric property"?

Definiion 1.5 Geometrical object is a set of points derived as joins and intersections

of elementary geometrical objects. What is that?. Finite intersections and unions? Maybe infinite ones?. It isn't clear. 

Definition 1.6... so on

I think you can find and compare your definitions with the Hilbert's axiomatization of the plane, for example in a book of Francis Borceux "An Axiomatic Approach to Geometry". Page 305.

Finally, a geometry is a theory (theory has a precise definition!) and we recognize it in his models: R^2, manifolds, graphs, etc.

Look the Hilbert axiomatization of the plane. You need, a set Pi, such his elements shall be called points and Delta, family of subset of Pi called lines. For some models, you need topology and sofisticated things, for example some flat manifolds. For some models, you need only elementary things, for example the plane. The same for others geometries.

The mathematicians did a lot of things! You have to investigate. I think, you can find solution for your exigencies. Else, you have to formulate exactly what you need, not only with empiric language.

Dear Jose and Horvath,

I should like to note, that I have referred to the paper “Geometry without geometry, whereas I wanted to refer to the paper "Geometry without topology as a new conception of geometry." Int. Jour. Mat. & Mat. Sci. 30, iss. 12, 733-760, (2002), (Available at http://arXiv.org/abs/math.MG/0103002 ). The first paper is a rough version. I have mixed up them. Now the physical geometry (T-geometry) is well developed. It is constructed as a result of a deformation of the proper Euclidean theory. It contains two steps:

(1) the proper Euclidean theory GE is presented in the sigma-representation, when GE is described in terms of the world function \sigmaE and only in terms    of \sigmaE. It is always possible. All geometrical objects of GE and all relations of GE are expressed in terms of \sigmaE and only in these terms

(2) The Euclidean world function \sigmaE is replaced by the world function \sigma of the generalized geometry G in all expressions of GE. As a result one obtains all geometrical objects and all geometrical relations of the generalized geometry G. Such a replacement means a deformation of GE.

After this deformation GE obtains some additional properties (for instance, multivariance). The proper Euclidean geometry may not have these properties.

Dear Mohamed, the question was, whether the definition of dimension can lead to indefinite result. But you answer that there are geometries, where the dimension is infinite. It is not answer to my question.

Dear Mohammed,

My question was: ‘Can geometry have no definite dimension?”. The (space-time) geometry G is defined completely by a metric \rho , given on a set of points \Omega. To determine dimension of a geometry G, one needs to give definition of dimension n of a geometry G in terms of the quantities \rho and \Omega and answer, whether or not the quantity n may be indefinite. If you like to express this in terms of fractals, topological spaces, Housdorff space, and other concepts, whose relation to the geometry G is not clear, then tell me, please, how these concepts can be expressed in terms of \rho and \Omega. Otherwise, your considerations are not an answer for my question.

Dear Mohamed,

At first, the authorities are important in music, but they have no relations to geometry. In mathematics one uses definitions instead of authorinies. In the Euclidean geometry the dimension is the maximal number n of linear independent vectors. The same definition is valid in any space-time geometry. In order that the dimension of a geometry exists, this maximal number n must be the same at all points of the space-time geometry.

Second, mathematicians like to turn simple things to very complicated ones. They invent a theory of dimension, Cantorian manifolds, and other clever things instead of giving a definition of the dimension and investigating, whether this definition gives a definite result in the real space-time geometry (but not in irreal Cantorian manifolds).

Yuri,

we can speak lot about many possible mathematics, but we can not physically  construct more than 5 regular polyhedra in our space. It is topological property of 3 dimensional space and it can be proved.

Dear Eugene,

Division of a geometry into: (1) geometry in itself, (2) topology, (3) coordinate system,  and  (4) dimension is absurdity. Geometry should be described by minimal number of fundamental concepts. The best case, if the geometry is described by the only fundamental concept. Such a concept is known. It is the world function \sigma. This concept may be used in continuous geometry and in the discrete geometry. All geometrical concepts and all geometrical objects can be described in terms and only in terms of the world function. For instance, let us consider geometry, where there is a minimal length \lambda. Mathematically it means that

|\rho(P,Q)| does not belong to (0,\lambda) for  all P,Q belonging to \Omega      (*)

where \Omega  is the set of points, where geometry is given. Usually, one considers (*) as a constraint on \Omega. As a result one obtains a geometry on a lattice. Geometry on a lattice is not a geometry. It is a caricature on geometry. To obtain a discrete geometry, one needs to consider (*) as a constraint on metric \rho or on the world function \sigma. If \Omega is a set of points, where the geometry of Minkowski is given, then the world function \sigmad of the discrete geometry Gd has the form

\sigmad = \sigmaM+\lambda2/2sgn(\sigmaM)    (**)

Where \sigmaM is the world function of the geometry of Minkowski . Now one can calculate dimension of Gd. The metric dimension is the maximal number of linear independent vectors One needs to calculate the Gram’s determinant    

   Fn(Pn) =||(P0Pi.P0Pk)||,  i,k.= 1,2,…n. Let us consider vectors P0P1=(l,0,0,0), P0P2=( 0,l,0,0),  P0P3=( 0,0,l,0),  P0P4 =( 0,,0,0,l), P0P5 = (a,0,0,0)

Calculation gives F4(P4)=l4+O(\lambda),  F5(P5) =\lamda+O(\lambda2),… See details in.  

 Metrical approach to geometry and discrete space-time geometry :

http://gasdyn-ipm.ipmnet.ru/~rylov/magdstg2e.pdf

Thus, topology is contained in the world function. Separate consideration of topology entangles the situation.

No separations, Yuri. Simply, we can consider topological properties without metric.

Dear Yuri,

a metric space (\Omega, \rho: \Omega \times \Omega \to [0, \infty) ) or equivalently (\Omega, \sigma = \rho^2)  is a topological space, and so, as I explain above, has well defined topological dimension provided every point has a neighborhood that is homeomorphic to a ball in R^n  for some n, which then happens to be the topological dimension [1]. In particular the metric defines the dimension.

There are of course examples where you want to use different notions of dimension.

In metric measure spaces (aka Polish spaces)  (\Omga, \rho, \mu), in particular in discrete metric spaces, you can consider the growth of the measure of balls as r \to 0 or r \ to \infty.  For a manifold the r \to 0 version gives the same dimension as the topological one.

For Noetherian spaces (e.g. the spectra of Noetherian rings or more general algebraic schemes in the sense of Grothendieck) we can define the dimension as the maximum length

X_0  \subsetne  X_1 \subsetne  X_2 ... \subsetne X_n

where each X_i is an irreducible closed set.

 But in any case, there is no mystery, and the notions are direct (but different) generalisations of the notion of Euclidean dimension suitable in different circumstances. 

[1] Strictly speaking that dimension can vary from to point if your space is not like a manifold, and has local models more complex than R^n for some n. The "correct generalisation of dimension for the very general class of finite dimensional CW complexes becomes the local homology sheaves of the dualising compex. For example, IIRC  a space like this:

-----

has different homology spaces of the dualising sheaf on the boundary of  the square , on the interior of the square, on the line ------, on the joining point of the line and the square and the endpoint of the line. So in a very real sense it is a space with no definite dimension. 

Dear Rogier,

As I have understood, we speak about different things. I am a physicist, and the geometry for me is the space-time geometry. In other words, geometry must describe space-time properties of physical bodies and physical phenomena. I qualify such a science as physical geometry. If one introduce temperature and flavor as geometrical properties, I  shall not agree to consider a science investigating these properties as a physical geometry. All geometrical properties are to be described in terms of the world function \sigma and only in terms of the world function \sigma. In the proper Euclidean geometry all geometrical properties can be described in terms of world function and only in terms of world function. The same is valid for other physical geometries, because any physical geometry G is obtained from the proper Euclidean geometry GE by means of a replacement of Euclidean world function \sigmaE by the world function \sigma of  physical geometry G. Dimension D of GE is determined as the maximal number of linear independent vectors in GE. 

Linear dependence of n vectors is determined by the Gram’s determinant Fn(Pn+1)=det||(P0Pi.P0Pk)|| ,  i,k=1,2,…n   Pn+1 ={P0,P1,…Pn} Scalar product (P0Pi.P0Pk) is defined via world function of points P0,Pi,Pk.

In order that the maximal number of linear independent vectors exist, the world function is to satisfy some constraints. If  this constraints are not satisfied, the dimension cannot be introduced. For instance, for a discrete space-time geometry the metrical dimension (the maximal number of linear independent vectors) cannot  exist.

As to topology, it can be taken into account by the world function. For instance, the topology of two-dimensional Euclidean plane and that of the cylinder obtained by identification of points with coordinates (x-L,y) and (x+L,y) is different. The world functions are also different

2\sigmap(x,y;x’y’)=(x-x’)2+(y-y’)2 and 2\sigmac(x,y;x’y’) = minn{(x-x’+2nL)2+(y-y’)2}  where n is integer and 2L  is the length of the circle, formed by the cylinder.  However, the metric tensor is the same in both cases.  It means, that in a physcal geometry, described in terms of the metric tensor,  one needs additionally a topology. Thus, the topology is determined by the world function, and one needs no other topology.

The case with other definition of topology does not used in the physical geometry.

Of course, mathematician may use another definition of  a topology. But I qualify geometry with such a topology as a mathematical geometry. The mathematical geometry is a mathematical toy, which is used for training of  the deduction capacities of mathematicians. The mathematical geometry turns to a real geometry, when and if one discovers some region of natural sciences, where such a geometry can be applied. 

Dear Eugene, 

See my previous answer to Rogier Brussee.

I understand your view point, Youri. But my view point is that geometry is mental feature by definition (You names it mathematical geometry) and I see no need to limit it by defined by You physical geometry. We can not prove what is more physical.

Dear Yuri,

you keep mentioning the world function but as you write somewhere in your posts, the world function \sigma is just the square of the metric \rho on the metric space (G, \rho). I think you are mixing up the Riemannian metric g in each point, with the metric \rho on G. Of course g defines \rho by 

\sigma(P,Q) = \rho^2(P, Q)  = inf_{\gamma, \gamma(0) = P, \gamma(1) = Q} (

                                       \int g(\dot \gamm, \dot \gamma) )

In fact what I try to point out (but seem to fail) is that the dimension of physical space is indeed completely determined by \rho because \rho determines the topology, and the topology determines the dimension. Moreover, this dimension is  the same as the linear dimension of the tangent space if space time can be given the structure of a Riemannian manifold.

Dear Rogier,

I am sorry, but I did not understand your post. What do mean \gamma, and  \gamma(0). Why do you think, that I mix metric tensor gik, with the metric as a distance between two ponts?. Does your term “metric space (G, \rho)”  contain the triangle axiom in its definition? What does mean (\dot \gamm, \dot \gamma)? Is it indices ?

Dear Yuri,

sorry, I should have explained my notation.  The infimum is taken over all differentiable pathes with endpoint P and Q, i.e. all differentiable maps

\gamma:[0,1] \to G with \gamma(0) = P , \gamma(1) = Q.

This gives a velocity vector v = v(t) = (\dot \gamma)(t) in the point \gamma(t), from which we can take the norm squared

g(\dot\gamma, \dot \gamma) = g(v, v) = g_{ab}v^a v^b

for every t, which we then integrate over t from 0 to 1. 

The reason I thought you are mixing up the metric tensor with the metric is because you say that the world function of the cylinder and the plane is different (which of course it is, and I never claimed otherwise) but  in local coordinates the metric tensor is the same, or more abstractly, the metric tensor on the flat plane R^2  is the pullback of the metric tensor on the cylinder R^2 / Z.

Finally, yes the term metric space (G, \rho)  does indeed contain the triangle inequality in its definition.  In fact in the usual mathematical notation a metric space (X,d) is a set X together with a function

         d: X \times X \to [0\infty)

such that

a) for all x, y \in X,  d(x, y) = d(y,x)  

b) for all x in X, y \in X , d(x,y) >= 0 (by defn.) and d(x,y) = 0 iff x = y.

c) for all x,y,z \in X,  d(x, z) \le  d(x,y) + d(y, z)

The metric then defines a topology on X by defining a set U to be open if every point x \in U there is an \epsilon > 0 such that the ball B(x, \epsilon) is contained in U.

This is all very standard but I thought I mention it for future reference and other readers.

Rogier,

and if metric (norm) is p-adic?

 The p-adic numbers are metric spaces In fact they are ultra metric spaces which means that (c) is replaced with the stronger 

c')  d(x,z) \le max (d(x,y), d(y,z))

(i just realised that in the post above I got the inequalty reversed, fixed that)

Dear Rogier,

We speak about different things. I  did not use the term metric space (or metric geometry). I wrote about physical geometry, which is defined as geometry G={\Omega, \sigma}, which is described completely by the world function \sigma. \Omega is a set of points, where the geometry is given. \Omega is an arbitrary set of points, and it is not necessary a manifold. The world function is defined by the relations

\sigma  is a real function on \Omega\times \Omega,   \sigma(P,Q) = \sigma(Q.P),  \sigma(P,P)=0  for all P,Q \in\Omega.

It is  all. There are neither triangle axiom, no positivity of distance d=\sqrt(2\sigma)

The physical geometry does not need to introduce additionally topology, I work with such a physical geometry. How to work with it? See, for instance, my paper   “Metrical conception of the space-time geometry”  Int. J. Theor, Phys.54, iss.1, 334-339, (2014), or “ Multivariance as immanent property of the space-time geometry”, Int. J. Theor. Phys 52,  iss.11, 4074-4082, ( 2013) 

Article Multivariance as Immanent Property of the Space-Time Geometry

Article Metrical Conception of the Space-Time Geometry