9.The Digital Topological Spaces

THIS PROJECT IS UNDER THE NEXT PHILOSOPHICAL PRINCIPLES

1) CONSCIOUSNESS IS INFINITE. CONVERSELY  THE INFINITE IS A FUNCTION AND PROPERTY OF THE CONSCIOUSNESSES.

2) BUT THE PHYSICAL MATERIAL  WORLD IS FINITE.

3) THEREFORE MATHEMATICAL MODELS IN THEIR ONTOLOGY SHOULD CONTAIN ONLY FINITE ENTITIES AND SHOULD NOT INVOLVE THE INFINITE. 

THIS PROJECT THEREFORE IS CREATING AGAIN THE BASIC OF MATHEMATICS AND ITS ONTOLOGY WITH NEW AXIOMS THAT DO NOT INVOLVE THE INFINITE AT ALL.

The topology in digital mathematics describes concepts of continuity and the axiomatic digital Euclidean geometry and digital differential and integral calculus.

The main concept here is not the open set but the point  seemingly in contact with a set A (in classical mathematics topology this corresponds to that the point a belongs to the closure of A, or that a net of points from A is converging to the  point a) . The axioms are not different . 

We define it as best example in Cartesian products of Digital real numbers (Higher Dimensional Digital analytic Euclidean Geometry)
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IN THE NEXT WE PREFER TO USE A 4-PRECISION LEVELS SYSTEM OF DIGITAL REAL NUMBERS. R(m,n,p,q) , where the measurements are done with the first two precision levels P(m), P(n), and the visible and invisible points are the next two precision levels P(p), P(q).

Creating 3D mental images with the imagination

The physiscists andd nobel prize winner R Feynamnn often was explaining that he was training his imagination imagining in 3D mental images the moving molecules of a gas in  box, and perceiving that temperature as their avearge velocity, the ressure as the average moemntum of the walls of the box, and the density as the average number of particles per unit of volume, He was explaing that he was getting much pleasure from this spiritual excerise of craeting 3D mental images and simulating the reality. Excatly this 3D VISUAL MENTAL IMAGES LANGUAGE parctice is  UTILIZED BY ME AND  is  required by someone to understand and develop the digital or natural mathematics, in geometry, real numbers and analysis.

Rules for phantasy and drawing of figures.

As initially we considered a system of digital real numbers R(m,n,p,q) we consider the points of P(m), P(n) as visible in the figures while the points of P(p) as invisble pixels , and those of  P(q) as invisible atoms. Therefore, even the points and seemingly infinitesimals that will be defined below, of P(n) relative to P(m) are considered as visible. This is in accordance with the habit in classical mathematics to make the points visible, although they claim that they have zero size.

The relation, a point is at seemingly infinitesimal distance from the points of a set or in seemingly contact with the set (of the digital real numbers) defines a digital topology. The point x is seemingly in contact with the set A of visible and invisible points of the R^s(m,n,p,q) (s-dimensional Cartesian product of digital real numbers) if and only if there is an invisible point y of A such that ||y-x||  (as norm of vectors) is s seemingly infinitesimal of the Low precision level, in other words  it belongs to  the high precision level but it is equal to zero to the low precision level. Τhe topological closure in the digital topology should have the idempotent property as in the classical analogue topology. In order to have the idempotent law of the closure of a set of points we need to use at least two kinds of points the phenomenological and the ontological or two precision levels or resolutions, invisible or high precision or resolution and visible or low precision or resolution defined by the Cartesian coordinates and their precision levels. So for every set A of visible and invisible  points there are two  operators ip(A) which is the set of invisible points of A and vp(A) which is the set of visible points of A.  We need not only the concept of  an invisible point seemingly contact with an invisible point, and visible point seemingly in contact with a visible point but also of an invisible point seemingly in contact with a visible point. Then the closure of a set of invisible and visible points is the original set plus the set of all visible (low precision)   points (without adding more invisible points  from their  contained invisible points but including the initial invisible points) that are seemingly in contact with any invisible point of the original set. We denote this closure of a set of invisible and visible points by cl(A).  

(Similar definitions of different closure operators can be given if we allow only invisible points).

Thus a visible point x is seemingly in contact with the set A, if and only if x belongs  to cl(A).


Being in a digital system of real numbers o 4-precision levels R(m,n,p,q) is somehow immaterial if we apply the definition of closure to visible or invisible points. But not so if we are in 3-precision levels R(m,n,q).

The closure operator is idempotent only if we repeat the closure only countably finite many times  (=not seemingly infinite many times in P(m)=uncountable finite many times )

The boundary Bd(A) of  a set of points are all the points that are seemingly in contact both with the set A  and its complement A^(c).

The interior of a set A is defined as the set int(A)=cl(A)-Bd(A).

And as usually the next relations holds Bd(A)=cl(A)-intersection-cl(A^(c))

And cl(A)=int(A)U Bd(A).  A set is open iff  A=cl(A^(c)). But also a set A is open iff  A=int(A).

A closed set of invisible and visible points is such that its closure gives the same set of visible points. Such a closure operator has the idempotent law of topology. And of course an open set is defined after the closure operator rather in the usual way: If its complement within the high resolution   and its closure is disjoint with the original set.

Let us  denote by cmplip(A) the complement of a set of invisible points in all invisible points and cmplvp(A) the complement of a set of visible points in all visible points.
Then a set is open if and only if  cl(cmplip(ip(A))) has no common points with A. 

This set of open sets has the properties of a usual topological space

1) the empty set and all the space are open
2) Any finite intersection of open sets is an open set
3) Any union of open sets is an open set

This topology describes the usual digital continuity of digital functions in any Cartesian dimension of the digital real numbers.

The boundary of a set is all the visible points that are in contact both with invisible points of the  set and with invisible points of the complement of the set in the high resolution.

A closed set obviously contains its boundary. And conversely a closed set  it can be defined equivalently as set of visible and invisible points that it contains its boundary too and the boundary of any subset of it.

In this topological way the continuity is define in the same way in all Cartesian product dimensions.

The concept of topological lowest visible compactness is defined in the usual way , where far the existence of finite subcover for any cover, we require , existence of lowest visibly finite cardinality of a subcover. Similarly for the concept of lower visible compactness or simply visibly compactness of a set of points.

See also post 10

DEFINITION  2.1

A digital real function defined on a closed interval f :[a,b]_m  ->P(m), of: :[a,b]_n  ->P(n), is (digitally) continuous at a point x of its domain of definition [a,b]_m in P(m), if and only if for every other point x’ of its domain of definition [a,b]_n   in P(n), such that x,x’ are of seemingly infinitesimally distance   dx=x’-x (belongs to P(n))  ,relative to P(m) , then also the dy=f(x’)-f(x) is  seemingly infinitesimally relative to P(m). In symbols:

dy=_ndof(x)=m dx=m 0

We concentrate on functions of P(n) of R(m,n,p.q) and we leave unused the precision levels P(p), P(q). We mention only that the definitions can be also for the pairs of precision levels P(m)/P(p), P(m)/P(q) as finer forms of differentiation and integration. If it is for all precision levels then it seems equivalent to the classical definitions.

RULE OF PHANTASY AND DRAWINGS:

NOTICE THAT AS P(n) IS ALSO CONSIDERED OF VISIBLE  PPOINTS THE SEEMINGLY INFINITESIMALS AS POINTS IN DERIVATIVE AND INTEGRATION  ARE CONSIDERED AS VISIBLE AS IS ASO CUSTOM IN CLASSICAL MATHEMATICS DRAWINGS. BUT HERE THERE ARE ALSO TRULLY INVISBLE POINTS THE PIXELSAND ATOMS.

Similarly we may define continuity by the pairs of precision levels P(m)-P(p), and P(m)-P(q).

If digital real function is digitally continuous at all points of its domain of definition it is called a (digitally) continuous digital real function.

The relation, a point is at seemingly infinitesimal distance from the points of a set or in seemingly contact with the set (of the digital real numbers) defines a digital topology. Similar to the concept of limit point of a set in a topological space. The topological space as classical topological space is initially defined fir the high precision P(high)=P(n) and denoted by oX and not the P(low)=P(m)+X.  Nevertheless the topological closure in the digital topology should have the idempotent property as in the classical analogue topology.

We consider the Cartesian product set P(Low)xP(HIgh) =P(m)xP(n), where we define the disjoint union  space  P(low)+P(high) and we do not consider that a coarse point contains fine points but we treat them separately. Our topological space will be the Y=X+oX. Subsets A of Y can be split to A=oA+vA , where oA are the fine points of A and cA are the coarse points of A.

DEFINITION 2.2

A point x of X=P(low)=P(m)   or of oX=P(high)=P(n)  is a limit point of  a subset  A of Y= P(low)+P(high) (and oA is a subset of oX) , iff there is a positive seemingly infinitesimal de of P(high) such that  for any positive seemingly infinitesimal da of P(high) less that de , there is a fine point y of oA such that |x-y|=da. We denote the set of fine of P(high) limit points of A by oL(A) and all coarse points of P(low) by L(A). We define as closure cl(A) of a subset A of Y , the cl(A)=A union Cl(A).

Notice that with the closure we add only coarse visible points not fine (possibly invisible)  points.  For this reason the closure operator has the idempotent low Cl(Cl(A))=Cl(A). In addition Cl(A union B)= Cl(A) union Cl(B) and Cl(A intersection B) =Cl(A) intersection Cl(B). We define that a x point of Y is seemingly in contact with the subset A of Y  iff x belongs to

Cl(A). In other words either it belongs to the set or it is a limit point of it.

The concepts of boundary points and interior points are defined so as to have the usual properties as well as the concept of open set,  base of open sets and base of neighbourhoods in Y.  Similarly for connectedness. Separation properties Tx might not be necessary at first approach.

At the end of this chapter, there is 

a) Advantages-disadvantages of these new digital mathematics compared to the classical analogue, infinitary mathematics. 

b) A fictional discussion in dialogue form of celebrated historic creators of the old mathematics praising or questioning the new mathematics compared to the old.