Geometric Continuum, Coordinatization and Hausdorff Spaces Kaushik Ghosh

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Kaushik Ghosh. Geometric Continuum, Coordinatization and Hausdorff Spaces. 2021. ￿hal-03092015￿

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HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Geometric Continuum, Coordinatization and Hausdorff Spaces

Kaushik Ghosh∗ Vivekananda College,University of Calcutta, 269, Diamond Harbour Road, Kolkata - 700063, India

Abstract

In this article, we will discuss the completeness of real numbers in the context of an alternate definition of the straight line as a geometric continuum. According to this definition, points are not regarded as the basic constituents of a line segment and a line segment is considered to be a fundamental geometric object. This definition is in particular suitable to coordinatize different points on the straight line preserving the order properties of real numbers. Geometrically fundamental nature of line segments are required in physical theories like the string theory. We will construct a new topology suitable for this alternate definition of the straight line as a geometric continuum. We can introduce a universal set of geometric continua of different dimensions including zero to construct topological spaces. The new definition of the straight line as a geometric continuum can be re-expressed as a connected topological space in the topology of geometric open intervals. This topology satisfies the normality condition and coincides with the standard topology of real numbers through coordinatization. We will discuss normality condition for these topologies of straight line and a few of its consequences.

MSC: 51P05, 54D10, 81T30

Key-words: geometric continuum, Hausdorff condition, string theory

1. Introduction

In this article, we will discuss an alternate definition of geometric continuum like the straight line that is more consistent with the linear continuum structure of real numbers [1,2,3]. Here a continuum need not to be compact [2]. The term straight line refers to the one dimensional line in geometry that can be coordinatized by the real numbers. The term infinite straight line corresponds to the extended real line. We will find that points need not to be the basic constituents of a line segment and a line segment can be considered to be a fundamental geometric object. This is consistent with microscopic physical theories where the elementary particles are considered to be extended objects without having any finer structure. One such theory is the string theory. We will reformulate the completeness axiom of real numbers that is more consistent with the new definition of straight line. We can generalize the above aspect to higher dimensional geometric continua. A geometric continuum of dimension n is not just a collection of geometric continua of dimension n − 1 and is a fundamental geometric object. This is important for the foundations of set theory and topology. We note that the intersection of two line segments like PQ and QR can be a point Q which is different from either set when we consider a line segment to be a fundamental geometric object. We also face problem to construct a topology. Considering the straight line only, we have two kinds of fundamental objects in it: geometrical points and line segments, and we have to construct open sets to have a convenient topology in the straight line. We solve this problem by assuming that the geometric continua of dimension one and zero form a universal set with a line segment being a geometric continuum of dimension one and a point being a geometric continuum of dimension zero. We can introduce familiar topologies in this universal set. The new definition of straight line as a geometric continuum is naturally expressed as a connected topological space in the topology given by geometric open intervals on the straight line. This topology coincides with a topology derived from a geometrical simple order relation present between different points on the straight line. This topology satisfy the Hausdorff and normality conditions [2] due to the fundamental nature of line segments and coincides with the standard topology of real numbers [2] through coordinatization. Note that to begin with there are now two kinds of ’points’ present in the topological space. In this article, we refer Hausdorff condition to mean separability of geometric points while normality indicates separability of

∗ E-mail [email protected] 2 disjoint line segments. It is more natural to rename normality condition as also the Hausdorff condition in the straight line and we will discuss this aspect in a future article.

2. Geometric Continuum, Coordinatization and Hausdorff Condition in the Straight Line

In this section, we will confine our attention on the straight line which is a geometric continuum of one dimension. We will give a precise definition of geometric continuum shortly. Henceforth, we will denote the straight line by the symbol E1 although we have not yet introduced any metric on the straight line. We now define a topology in E1. For any two distinct points a, b on E1, we can define a geometrical simple order relation between them that asserts that a < b if b lies at the right of a. We choose a basis element as (a, b): the open interval defined as the entire line interval between a and b excluding themselves with a < b in the geometrical simple order relation. We get the null set when a and b coincide. We get a topology on E1 by adding the null set to the basis elements obtained above. We denote this topology by U. The line segment ab, also denoted by [a, b], is the the entire line interval between a and b that includes a and b. We can generate different basis elements by considering a, b to be a pair of indices that take values from the same index set K. K should be large enough to give all possible geometric open intervals on E1. It is not a trivial issue that (a, b) can be expressed as an open interval from the set of real numbers R which fixes K = R = (−∞, ∞). This is done through the completeness axiom of Dedekind which states that for any point on the number axis, a straight line, there corresponds a real number from R [4,5]. The real numbers include the rational and irrational numbers. We can then coordinatize different points on E1 when we choose a suitable point on E1 to be represented by zero and a particular point to be represented by one. The completeness axiom and the algebraic properties of real numbers as a field leads us to introduce the Euclidean metric on E1 [2]. The distance between two points with coordinates x, y is given by |x − y|. Existence of coordinatization indicates that E1 can inherit a simple order relation present in R with the following additional properties [2]:

O1. If x(a) < x(c) there is a point b between a and c such that x(a) < x(b) and x(b) < x(c),

O2. For any nonempty A ⊂ E1, there is a least upper bound for x(A) when x(A) is bounded above.

Here x(a) denotes the coordinate of the point a ∈ E1 and x(A) includes the coordinate for any point that belongs to the subset A of E1. The order property of R introduce an order topology in E1. In E1, the topology U can be now given by (x(a), x(b)). This topology and the order topology inherited from R coincide when we choose x(a) < x(b) [2]. This topology also coincides with the standard topology of R given by the open sets (x, y), x, y ∈ R and x < y. Note that we can use (x(a), x(b)) to denote open balls in the Euclidean metric topology through the midpoint property of Euclidean metric in the straight line. (x(a), x(b)) gives us an open ball centered at the midpoint of the interval. The linear order property O1 brings us closer to an alternate definition of E1 as a geometric continuum through the following axiom [1]:

AXIOM 2-1. Points are not the basic constituents of a straight line. A line segment is not just a collection of points and is a fundamental geometric object.

The above axiom may be apparently counter intuitive since two straight lines intersect at a point. How- ever, whatever may be the cardinality, a collection of zero-extension points can not give an extended and homogeneous object like a one dimensional straight line. The same is obviously valid for line segments. Also a homogeneous collection of points give us a single point since the extension of a point is zero. We can illustrate the last aspect in the following way:

We consider a homogeneous linear array of marbles touching each other. If we now shrink the volume of every marble keeping them in contact (so that the array is always homogeneous), we will get a single point when the volume of every marble is made to vanish.

Axiom 2-1 is consistent with the order property O1. We do not need adjacent points to construct a line. The notion of adjacent points is required not to have gaps in the straight line when we consider it to be merely a collection of points [6]. This contradicts O1 [6]. Axiom 2-1 allows us to impose the order property O1 on the geometrical simple order relation in E1 mentioned at the beginning of the present section where a < b if b lies at the right of a. Every pair of different points on E1 are now separated by an open line 3 interval, and we always have a point in between any such pair. Thus the geometrical simple order relation satisfies the order property O1. It is not difficult to impose the order property O2 on the geometrical simple order relation present in E1. We can introduce the symbol ∞ to represent unboundedness of E1 in this order relation. E1 then includes all points e with −∞ < e < ∞. This indicates that E1 is unbounded in both directions similar to the real numbers. We can impose O2 to any proper subset of E1 bounded in the geometric order relation. A few physical aspects of the defining Axiom 2-1 of geometric continuum is mentioned in [1]. Here we note that Axiom 2-1 is required in the string theory of theoretical physics [7]. In string theory, point particles are replaced by one dimensional strings as the most elementary building blocks of matter and radiation. We can not assume a string to be a collection of points, in which case it will not be a fundamental object and we will have to hypothesize a new interaction that will hold the points forming a string. Since string theory involves high energy physics of quantum gravity, it is unlikely that we will have such an interaction that will always bind the points forming a string at the high energy scale of quantum gravity. Similar discussions hold if we assume that elementary particles like electrons or quarks are not point particles and have a radius at least greater than their Schwarzschild radius. The Schwarzschild radius of an electron is approximately 1.35 × 10−51m. Note that black holes do not have baryonic or leptonic numbers by the no-hair theorems [8], and it is unlikely that an electron has a leptonic number as well as a radius smaller than its Schwarzschild radius. We can generalize Axiom 2-1 to higher dimensional geometric continua. A geometric continuum of dimen- sion n is not just a collection of geometric continua of dimension n−1 and is a fundamental geometric object. This is important for the foundations of set theory and topology. We note that the intersection of two line segments [a, b] and [b, c] is a point {b} which is different from either set when we consider a line segment to be a fundamental geometric object. We also face problems to construct a topology. We now have two kinds of fundamental objects in E1: geometric points and line segments. We want to use open line intervals to construct a simple topology in E1 [9]. We remove this problem by adapting the naive definition of sets: sets are collections of objects which are closed under the operations of set theoretic union and intersection. This definition allows us to introduce the null set. Thus, we assume that geometric continua of different dimensions form a universal set with a point being a geometric continuum of dimension zero. We represent this universal set by the symbol M . We confine our attention to only E1 which is a subset of M and try to construct a topology for itself. It is now possible to define set theoretic operations involving line segments and points since they are subsets of the same universal set. We construct a topology similar to U by defining (a, b) as [a, b] − {a} − {b} = [a, b] T {a}c T {b}c. We represent this topology by a different symbol V, since (a, b) is no longer a mere collection of points. (a, b) is not also a collection of line segments and points. It is introduced to give a topology. We can not introduce (a, b) as a set without defining M . We need not to introduce (a, b) to do pure geometry. We can also use geometric points and line segments to give a topology [9]. We now derive a topology from the geometrical simple order relation. The basis elements of this topology consist of (1) the null set and the straight line, (2) for each point a on the line, the whole line interval at the left of a excluding a, (3) open intervals (a, b) for every pair of distinct points a, b on the line with a < b and (4) for every point b on the line, the whole line interval at the right of b excluding b. Again we note that, we have used the geometric order relation between any pair of distinct points on E1 to construct open sets that are not merely collections of points. We denote this topology by D. In this topology, we can not separate the straight line into two mutually disjoint open sets. Thus, Axiom 2-1 is consistent with the formal definition of connected topological spaces in the topology D. This topology coincides with the topology of geometric open intervals V. We can represent E1 as (−∞, ∞) in the topology V. In V also, we can not separate E1 into two disjoint open sets A and B such that E1 = A S B. We can consider the above conclusions as topological definition of the straight line as a geometric continuum that follows from Axiom 2-1. We now proceed to construct a representation of the index set K that is consistent with Axiom 2-1. With Axiom 2-1 as the definition of a line segment, we can reformulate the axiom of completeness of the real numbers as follows:

AXIOM 2-2. Any segment on the straight line E1 can be ascribed a real number greater than zero. The number can be either rational or irrational. We call this number to be the length of the segment. A point has zero length.

Note that, Axiom 2-2 refers to segments in E1 and is not stated in terms of points. We assign the same length to [a, b], (a, b], [a, b) and (a, b). The length of a line segment is uniquely determined provided we choose a given line segment to be of unit length. We now choose a suitable point on E1 as the origin and coordinatize any point on E1 according to the length of the line segment joining the two with a particular line 4 segment chosen to be of unit length. Points on the right side of the origin are assigned positive coordinates while those on the left have negative coordinates. With both E1 and R being unbounded in their respective order relations, we can assume K = R, where K is the index set mentioned before. We recover the standard topology of R when a, b in (a, b) are replaced by the coordinates of a and b given by x(a) and x(b) respectively. (x(a), x(b)) now represents the open line interval (a, b) on E1. As before, we can consider it to be an open ball in the Euclidean metric topology. However, we no longer consider it to be merely a collection of geometric points. Thus, we can assume that the topology D derived from the geometric order relation present in E1, the topology V of geometric open intervals in E1, the order topology in R and the standard topology in R coincide through Axioms 2-1 and 2-2. Axiom 2-1 and the order property O1 of R give us a homeomorphism between E1 and R through coordinatization. However, in this article we will continue to use different symbols for them to distinguish between geometry and algebra respectively. 1 It easily follows from Axiom 2-1 that E is Hausdorff (T2) and normal (T4) in the topology V given by the geometric open intervals (a, b). In this article, we refer Housdorff condition to mean separability of geometric points by disjoint open sets while normality indicates separability of disjoint line segments by disjoint open sets. Since line segments are fundamental objects, every pair of different points on E1 are always separated by an open line interval. They are contained within an open line interval and we can construct two disjoint open sets separating the pair by deleting any point within the line segment joining the two. The same is valid for two disjoint line segments PQ and RS with P < Q < R < S in the geometric order relation. PQ and RS are closed sets and are contained within an open line interval. The points Q and R are always separated by an open line interval and we can construct two disjoint open sets separating PQ and RS by deleting any point in the line segment joining Q and R. Thus, E1 is Hausdorff and normal in the topology V. However, it would be more appropriate to say that E1 is a Hausdorff space if we consider the fundamental nature of geometric points and line segments. We do not follow this approach for the present article. In a coordinatization scheme, the Housdorff and normality conditions are ensured by the completeness of real numbers and the order property O1 of R. Given a pair of different points a and c of E1 and a coordinatization where x(a) < x(c), O1 indicates that for any point b in between a and c there exists a number x(b) that coordinatize b such that x(a) < x(b) < x(c). (x(a−), x(b)), (x(b), x(c+)) with x(a−) < x(a) and x(c) < x(c+) are a pair of open sets that separate a and c respectively. Similar arguments can be used to establish normality condition by considering the coordinates x(Q) and x(R) of PQ and RS. This is expected since (a, b) and (x(a), x(b)) give the same topology as mentioned before. However, the Hausdorff condition in the topology U does not remain valid for every pair of points if we regard the straight line to be merely a collection of points. Look at the discussions given below Axiom 2-1 with reference to [6]. If we 1 assume the existence of adjacent points e1, e2 on E , the sets (u1, e2) and (e1, u2) with u1 < e1 < e2 < u2, that separate e1 and e2 no longer remain open in U. There is no open set containing e1 that is contained in 1 (u1, e2). Similar aspect remains valid for e2 and (e1, u2). With this definition, E is Hausdorff only in the discrete topology. Lastly, we note that x(a), x(b) in (x(a), x(b)) can be rational or irrational. We can not use the basis of open balls with rational radii centered at points with√ rational coordinates to give a Hausdorff topology in E1. This is due to the existence of irrationals like 2, π. The points with irrational coordinates should belong to one or other of these basis elements and hence can not be separated from all points with rational coordinates by open sets obtained from this covering. This is important for differential geometry. We can not have a topology in a differentiable manifold that is both Hausdorff and second countable.

3. Conclusion

In this article we have considered a new definition of geometric continuum where a geometric continuum of dimension n is not just a collection of geometric continuum of dimension n − 1 and is a fundamental geometric object. We have discussed this with reference to the straight line. A point is considered to be a geometric continuum of dimension zero. We have introduced a universal set containing geometric continua of different dimensions including zero to define topology in geometric continua. When restricted to the straight line only, the fundamental nature of line segments leads to a normal topology where the open sets are geometric open intervals. The standard topology of real numbers is same as the above topology through Axiom 2-2. The new definition of a geometric continuum is required in physical theories like the string theory. Further discussions with regard to cardinality of the rationals is communicated as a separate article and can be found in [10]. 5

References

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