University of Illinois at Urbana-Champaign

Department of Mathematics

## Closeness Spaces:

### Elementary Explorations Into Generalized Metric Spaces, and Ordered Fields Derived From Them

Jonathan Hayward

Master’s Thesis

Advisor: John Gray

May 6, 1998

## Abstract

A generalization of metric spaces is examined, in which we are able to determine which of two pairs of points is closer (or if both are equally close), but not initially know how to assign a number to a distance. After the spaces are defined in general, we look at some more specific closeness spaces, and establish the the existence of a metric, which we are able to determine, under certain broad circumstances.

After looking at the closeness spaces, more specific attention is devoted to the closenesses themselves. We begin to define arithmetic operations over closeness spaces, and (given certain restrictions on the space) then complete addition and subtraction to develop a totally ordered group in which the closenesses are embedded. We prove that it is indeed a totally ordered field, and look at some examples. Directions are suggested for future research.

[*Side note when entering this dissertation two decades later: this research includes a way to rigorously define and use infinitesimals. Infinitesimals were long seen as something you wanted to have but could not rigorously define; epsilon-delta proofs in relation to derivatives in calculus represent a masterstroke of how to do an infinitesimal’s job using only standard real numbers for epsilons and deltas. Infinitesimals were spoken of as a ghost to be exorcized, and the entire point epsilon-delta proofs were a way to circumvent obvious use of infinitesimals in a mathematically rigorous way. At the time this thesis was written there appear to have been rigorous treatment of infinitesimals; however, so far as one could tell this approach to providing this kind of squeaky-clean rigorous handling of infinitesimals was new when the thesis was written.*]

## Introduction

Intuitively, a closeness space is like a metric space, with balls, symmetry, positive definiteness, and a triangle inequality, boundaries, an induced topology, and other familiar attributes of a metric space. However, it is a space for which we do not specifically know a metric: it is possible that we simply do not know a metric or none is given, or that no metric may exist. The latter holds in certain cases where the real numbers are too coarse of an ordered field to describe the space’s distances: such a thing is possible, for instance, when there are infinitesimal and infinite distances. A closeness space might not be thought of so much as a generalization of a metric space (at least in the sense that a metric space is a generalization of ℝ^{n}), but rather as a metric space with a generalization of real-valued distances. It is a metric space which may potentially have nonstandard real numbers (broadly defined) as its distances, rather than necessarily having real numbers under the standard model as its distances.]

In this sense, what is of interest is not only the spaces themselves, but their distances: what kind of group embeds them (we will look at a field which embeds an arithmetic closure of these distances). We will study the topological spaces, but our interest is not only in the spaces, but in the ordered groups and then fields which embed the closenesses. Throughout this thesis, the aim is both to establish certain elementary properties — laying a groundwork — and also to suggest directions for future research.

It is remarked that the approach is not to start with a field and then see for what kind of spaces it can function like a metric; the approach is rather to start with a space and see what kind of field acts as an arithmetic closure to its closenesses, given a certain construction.

## Chapter 1: Notation, Definitions and Terminology

### Notation 1:1:

In this document, a lowercase, indexed variable name is generally understood to be an element of the set designated by the corresponding uppercase letter, provided that the letter is ‘s’ or occurs after ‘s’ in the alphabet. For example:

*s _{1}* ∈

*S*

Furthermore, we associate in the same way *α* with indexing set *J*, and *β* with *K*. These indexing sets are understood to have no last element.

There will be plainly marked exceptions to this rule.

### Definition 1:2:

A *closeness space C* is a set *S*, together with a function

*ƒ*: *S* ✕ *S* ✕ *S* ✕ *S* ↦ {‘<‘, ‘=’, ‘>’}

such that the following conditions hold:

### Definition 1.2.1:

*f* is defined for each quadruplet of points in *S*.

(*S* is said to be a *space*, and its elements are referred to as its *points*, as elsewhere in topology. The function *f* is said to be a *closeness*.)

Intuitively, this condition and those following guarantee that *ƒ* is comparing the distance between the first two points, and the distance between the last two points to see which one is greater. This condition, and the next four, are simply conditions which guarantee that *ƒ* is well-behaved as a function on a pair of pairs of points, only depends on which pair of pairs of points is given, and defines a total ordering up to equivalence classes.

For every six points *s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}, *s*_{5}, and *s*6 (possibly non-distinct), we have the following conditions hold:

### Condition 1.2.2:

*ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = *ƒ*(*s*_{2}, *s*_{1}, *s*_{3}, *s*_{4})

*ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = *ƒ*(*s*_{3}, *s*_{4}, *1*_{1}, *s*_{3})

*ƒ* is not affected by swapping the elements in one pari, or by swapping the pairs. This is the closeness space’s version of a metric space requiring symmetry.

### Condition 1.2.3:

If *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘<‘, then *ƒ*(*s*_{3}, *s*_{4}, *s*_{1}, *s*_{2}) = ‘>’.

If *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘=’, then *ƒ*(*s*_{3}, *s*_{4}, *s*_{1}, *s*_{2}) = ‘=’.

If *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘>’, then *ƒ*(*s*_{3}, *s*_{4}, *s*_{1}, *s*_{2}) = ‘<‘.

### Condition 1.2.4:

If *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘<‘ and *ƒ*(*s*_{3}, *s*_{4}, *s*_{5}, *s*_{6}) = ‘<‘, then *ƒ*(*s*_{1}, *s*_{2}, *s*_{5}, *s*_{6}) = ‘<‘.

### Condition 1.2.5:

We have

*ƒ*(*s*_{1}, *s*_{2}, *s*_{1}, *s*_{2}) = ‘=’

### Condition 1.2.6:

If *s*_{2} and *s*_{3} are distinct, then we have

*ƒ*(*s*_{1}, *s*_{1}, *s*_{2}, *s*_{3}) = ‘<‘

Every point is closer to itself than the distance between any pair of distinct points; this is the closeness space’s version of the positive definiteness of a metric space.

### Condition 1.2.7:

We have

*ƒ*(*s*_{1}, *s*_{1}, *s*_{2}, *s*_{2}) = ‘=’

In other words, there is only one zero. It may be mathematically interesting to remove this restriction, but we will not investigate that possibility.

### Condition 1.2.8:

If *ƒ*(*s*_{1}, *s*_{3}, *s*_{1}, *s*_{2}) = ‘<‘ then for every set *T* ⊂ *S* containing points arbitrarily close to *s*_{3} (in a sense to be defined below), there exists *t*_{1} ≠ *s*_{3}, such that, for every point *s*_{4}, if *ƒ*(*s*_{3}, *s*_{4}, *s*_{1}, *s*_{2}) = ‘<‘.

(What this is getting at, is that if you have a boundary point *s*_{2} to a ball (boundary being outside the ball as with metric spaces), then every point closer to the center than the boundary point has a neighborhood entirely contained inside the ball (closer to the center than the boundary point). This means that a ball with a boundary point has a unique radius: there cannot be a second boundary point further than the center than the first boundary point, because then the first boundary point would be inside the ball; there also cannot be a secondary point closer to the center than the first boundary point, because this axiom says that every closer point has a neighborhood.)

### Condition 1.2.9:

A set *T* ⊂ *S* is said to hold points *arbitrarily close* to point *s*_{3} (in a sense to be defined below), there exists *t*_{1} ≠ *s*_{3}, such that if *ƒ*(*s*_{2}, *t*_{1}, *s*_{2}, *s*_{4}) = ‘<‘, then *ƒ*(*s*_{1}, *s*_{4}, *s*_{1}, *s*_{2}) = ‘>’.

(Here, we say that if you have a boundary point *s*_{2} to a ball, then every point further from the center than the boundary point has a neighborhood disjoint from the ball. Note that these two conditions may be vacuously satisfied by finite or other discrete metric / closeness spaces, with which we are not very much concerned.)

These two stipulations together constitute the closeness space’s version of the triangle inequality in a metric space. The slight awkwardness of this definition is necessary to permit discrete metric spaces. This awkwardness will recur in other places where we are defining concepts on a very low level without using familiar tools (because we are developing a more general form of such tools), but it should pass.

### Definition 1:3:

A set *T* ⊂ *S* is said to contain points *arbitrarily close* to point *s*_{1} if the following conditions hold:

### Condition 1.3.1:

*T* is nonempty and contains at least one point distinct from *s*_{1}.

### Condition 1.3.2:

For every distinct pair of points *s*_{2} and *s*_{3, there exists T1 distinct from s1 so that ƒ(s1, t1, s2, s3) = ‘<‘.}

(In other words, for every closeness in the space, there is a point in *T* that is closer to *s*_{1}.)

### Additional terminology 1.4:

#### Term 1.4.1:

Point *s*_{1} is said to be *closer to s _{2} than s_{3} is* (close to

*s*

_{2}) when

*ƒ*(

*s*

_{2},

*t*

_{1},

*s*

_{2},

*s*

_{4}) = ‘<‘.

#### Term 1.4.2:

Points *s*_{1} and *r* are said to be *equidistant from s _{2}* when

*ƒ*(

*s*

_{1},

*s*

_{2},

*r*,

*s*

_{2}) = ‘=’.

#### Term 1.4.3:

Point *s*_{1} is said to be *father from s _{2} than r is* when

*ƒ*(

*s*

_{1},

*s*

_{2},

*r*,

*s*

_{2}) = ‘>’.

#### Term 1.4.4:

A pair of points is referred to as a *distance*.

#### Term 1.4.5:

The pair (*s*_{1}, *s*_{2}) is said to be the *distance from s _{1} to s_{2}*.

#### Condition 1.4.6:

If distance *d*_{1} is the pair (*s*_{1}, *s*_{2}) and distance *d*_{2} is the pair (*s*_{3}, *s*_{4}), then the following three conditions hold:

##### Condition 1.4.6.1:

If *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘<‘, then *d*_{1} is said to be *less than d _{2}*, written

*d*

_{1}<

*d*

_{2}.

##### Condition 1.4.6.2:

If *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘=’, then *d*_{1} is said to be *equal to d _{2}*, written

*d*

_{1}=

*d*

_{2}.

##### Condition 1.4.6.3:

If *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘>’, then *d*_{1} is said to be *greater than d _{2}*, written

*d*

_{1}>

*d*

_{2}.

#### Remark 1:5:

Equality induces a partition of equivalence on distance. We will abuse notation slightly by referring to a distance, its equivalence class, and elements of its equivalence class interchangeably. Context should make clear which of these is meant; if context is not sufficient to clarify, then we will be more explicit as to which of these is intended.

### Definition 1.6:

A *ball about point s _{1}* is a set of points such that the following two conditions hold:

#### Condition 1.6.1:

Every point in the ball is closer to *s*_{1} than is every point not in the ball.

#### Condition 1.6.2:

There does not exist point *s*_{2} in the ball such that the following conditions hold:

##### Condition 1.6.2.1:

No point in the ball is further from *s*_{1} than *s*_{2} is.

##### Condition 1.6.2.2:

*S* contains points arbitrarily lose to *s*_{2}, which are not contained in *B*.

This latter condition guarantees that *B* does not contain its boundary, if it does have a nonempty boundary.

As the remainder of the definition and terminology, we have:

#### Condition 1.6.3:

If distance *d* = (*s*_{1}, *r*), *r* is not contained in *B*, and *B* contains points arbitrarily close to *r*, then ball *B* is said to of *radius d*, or to be the *ball of radius d centered at p*, and its boundary is said to be the *circle of radius d centered at p*.

(By the remarks following the triangle inequality, there is at most one equivalence class of distances which satisfy this property. Note that a ball might or might not necessarily have a radius.)

## Definition 1.7:

A set *T* is said to have points *arbitrarily close to* point *s*_{1} if *T* contains at least one point *t*_{1} ≠ *s*_{1}, and for every *t*_{2} ≠ *s*_{1}, there exists *t*_{3} which is closer to *s*_{1} than is *t*_{2}.

## Definition 1.8:

The *boundary* of a set *T* is the set *U* of points *u* such that both *T* contains points arbitrarily close to *u*, and *S*∖*T* contain points arbitrarily close to *u*.

## Definition of values, having different levels, 1.9:

We are using the term *value* to refer to mathematical objects which we will use in the construction of the field we are working on. Each value has a level; values of higher levels are determined in terms of values of lower level. The highest level of value will be an element of a field. I will define some (not all) levels of values here. If we use the term without specifying its level, it should be understood to be the last level specified, usually the highest level so far defined, if there is ambiguity. In some cases we will leave an ambiguity when what we are saying applies both to a member of an equivalence class, and its class.

### Definition 1.9.1:

A *level 0 value* is defined to be a distance (strictly defined as a pair of points).

### Definition 1.9.2:

A *level 1 value* is defined to be an equivalence class of level 0 values under the partition induced by equality. Level 1 values are ordered, in the same way that their members are ordered.

The remaining levels of values will be defined after I have begun to build up the the machinery necessary to explain and use them.

### Definition 1.10:

A *level 0 zero* is defined to be a distance (*s*_{1}, *s*_{1}).

### Definition 1.11:

A *level 1 zero* is defined to be the equivalence class of level 0 zeroes. Zeroes will be defined for all levels greater than or equal to level a.

### Definition 1.12:

A level n value is defined to be *positive* if it is greater than the level n zero.

### Definition 1.13:

A *level n sequence* is defined to be a level n sequence {∈_{α}}_{α∈J}, with *J* a totally ordered indexing set.

A *level n epsilon* is defined to be a level n sequence {∈_{α}}_{α∈J}, of positive level *n* values, such that the following two conditions hold:

#### Condition 1.13.1:

Every positive level *n* value *v* is greater than some ε_{α}.

#### Condition 1.13.2:

Every ε_{α} is greater than or equal to every ε_{β}.

#### Condition 1.13.3:

Distance *d*_{1} is said to be *within distance d2 of distance d3* if there is a set of points *s*_{1}, *s*_{2}, and *s*_{3} such that *d*_{1} = (*s*_{1}, *s*_{2}), * d2 ≤ ( s_{3}), and d_{3} = (s_{1, ss}*

_{3}).

## Chapter 2: Examples

### Example 2.1:

Every metric space is a closeness space. Two distances are compared by ‘<‘.

### Example 2.2:

We derive a space *C* from ℝ^{2} under the Euclidean metric as follows:

We make *C* a copy of ℝ^{2}, and then we add a point *o*‘ to the space, and define closenesses as follows:

*O*‘ is as close to every non-origin point as the origin is.

*O*‘ and the origin are closer than any other distinct pair of points.

#### Theorem 2.1.1:

*This closeness space cannot be described by any metric.*

##### Proof:

We prove this by contradiction.

Assume that such a metric exists.

If a metric did induce this closeness, it would have a least nonzero distance *d*, the distance from the origin *O* to *O*‘.

Let the distance from *O* to (0, 1) be *d*‘.

By the Archimedean property, there exists *n* such that *d*‘ ÷ *n* < *d*.

By repeated application of the triangle inequality on segments from (0, 0) to (*i* ÷ *n*, 0) and from (*i* ÷ *n*, 0) to ((*i* + 1) ÷ n, 0), this means that *d*‘ is at most equal to *n* times the distance from (0, 0) to (1 ÷ *n*, 0).

This means that the distance from (0, 0) to (1 ÷ *n*, 0) is less than *d*, but it is positive because they are two distinct points, and *d* is a minimal positive distance. ⇒⇐

Q.E.D.

This space is in many ways a space very like a metric space; although it boasts unusual decoration, it has a strong amount of stricture, like a metric space, structure that might not be present in an arbitrary metric space.

### Example 2.2:

Let *M* be a metric space with metric μ over a set *E* of equivalence classes partitioning a set *S*. Then we can define a closeness space *C* which has *S* as its space, and its closeness *ƒ* defined as follows:

For every four points *s*_{1}, *s*_{2}, *s*_{3}, *s*_{4} in *S*:

#### Case 2.2.1:

If μ(*s*_{1}, *s*_{2}) < μ(*s*_{3}, *s*_{4}), then *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘<‘.

#### Case 2.2.2:

If μ(*s*_{1}, *s*_{2}) > μ(*s*_{3}, *s*_{4}), then *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘>’.

#### Case 2.2.3:

If μ(*s*_{1}, *s*_{2}) = μ(*s*_{3}, *s*_{4}), then:

##### Case 2.2.3.1

If *s*_{1} = *s*_{2} and *s*_{3} = *s*_{4} then *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘=’.

##### Case 2.2.3.2

If *s*_{1} ≠ *s*_{2} and *s*_{3} = *s*_{4} then *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘>’.

##### Case 2.2.3.3

If *s*_{1} = *s*_{2} and *s*_{3} ≠ *s*_{4} then *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘<‘.

##### Case 2.2.3.4

If *s*_{1} ≠ *s*_{2} and *s*_{3} ≠ *s*_{4} then *ƒ*(*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}) = ‘=’.

In other words, if we have a metric space over equivalence classes, we can compare distances between pairs of elements of the classes by first looking at the distance between the elements’ equivalence classes, and then doing something else to break ties — say, seeing where they are the same.

In relation to this, we have:

### Definition and example 2.3:

If we have two closeness spaces *C* and *D*, with underlying sets *S* and *T*, then we can take their cross product *E* = *C* ✕ *D*, with underlying sets *S* and *T*, then we can take their product *E* = *C* ✕ *D*, with closenesses compared in the dictionary order.

Specifically, let *U* be the underlying set for *E*. We compare two distances *d*_{1} = (*u*_{1}, *u*_{2}) and *d*_{2} = (*u*_{3}, *u*_{4}), with *u*_{1} = (*s*_{1}, *t*_{1}) and *d*_{2} = (*u*_{2}, *u*_{2}), *u*_{3} = (*s*_{3}, *s*_{3}), and *u*_{4} = (*s*_{4}, *t*_{4}), as follows:

#### Case 2.3.1:

If (*s*_{1}, *s*_{2}) < (*s*_{3}, *s*_{4}), then (*u*_{1}, *u*_{2}) < (*u*_{3}, *u*_{4}).

#### Case 2.3.2:

If (*s*_{1}, *s*_{2}) > (*s*_{3}, *s*_{4}), then (*u*_{1}, *u*_{2}) > (*u*_{3}, *u*_{4}).

#### Case 2.3.3:

If (*s*_{1}, *s*_{2}) = (*s*_{3}, *s*_{4}), then:

##### Case 2.3.3.1:

If (*t*_{1}, *t*_{2}) > (*t*_{3}, *t*_{4}), then (*u*_{1}, *u*_{2}) > (*t*_{3}, _{4}).

##### Case 2.3.3.2:

If (*t*_{1}, *t*_{2}) < (*t*_{3}, *t*_{4}), then (*u*_{1}, *u*_{2}) < (*t*_{3}, _{4}).

##### Case 2.3.3.3:

If (*t*_{1}, *t*_{2}) = (*t*_{3}, *t*_{4}), then (*u*_{1}, *u*_{2}) = (*t*_{3}, _{4}).

*N.B. This cross product, in the dictionary order, will be used later.*

#### Example 2.3.4:

Let *S*, *T* = ℝ^{2} under the closeness induced by the Euclidean metric. Then *U* = *S* ✕ *T* may be described as a Euclidean plane, where each point is itself a miniature Euclidean plane. It is a plane with infinitesimal distances, or alternately an infinitesimal Euclidean plane.

A typical ball in this space is the ball with center at the origin ((0, 0), (0, 0)) consisting of all points strictly closer to the origin than ((1, 1), (1, 1)). This divides teh large-scale plane into three regions: the interior, exterior, and boundary of the disk of radius 1, centered at the origin. The interior of the disk corresponds to miniature planes which are entirely within the ball, where every point is inside. The exterior of the disk corresponds to miniature planes which are entirely outside the ball, where no point is inside. The boundary of the disk corresponds to miniature planes where the interior of the disk of radius 1 centered at the origin (of the small one, not the large plane or metric space) is inside the ball, and its boundary and exterior are outside. The boundary of the given ball in *U* consists of, in the miniature planes, all circles of radius 1 centered at the origin which are themselves on the circle of radius 1 in the large plane.

Proof that this satisfies the axioms of the space:

The set of equivalence classes (under equality) of closenesses has a 1-1 order-preserving mapping to the nonnegative real number line cross itself, in the dictionary order. In other words, it is a dictionary order cross product of two totally ordered sets, and therefore totally orders. This satisfies axioms 1.2.1-1.2.7.

To satisfy 1.2.8, we let *s*_{2} be closer to *s*_{3} than is *s*_{3}.

If the small planes of *s*_{2} and *s*_{3} are equidistant to the small plane of *s*_{1}, then the small plane position position of *s*_{2} is closer to the small plane position of *s*_{1} than is the small plane position of *s*_{3}. There is, by topology, an open disk about the small plane position of *s*_{1} and boundary the small plane position of *s*_{3}; if we take such a disk in the small plane *s*_{3} is actually contained in, it has points arbitrarily close to *s*_{3}, and is contained in the disk of center *s*_{1} and boundary *s*_{2}.

Every set *T* containing points arbitrarily close to *s*_{3} intersects the aforementioned disk infinitely many times. So we take some point inside that as our *t*_{1; every point s4 closer to se than is t1 and therefore closer to s1 than is s2.}

The same argument holds in the case that the small plane of *s*_{3} is closer to the small plane of *s*_{1} than is the small plane of *s*_{2}, save that we simply choose any disk contained in the small plane of *s*_{3}.

1.2.9 is satisfied; we simply have an open disk outside the open disk of center *s*_{1} and boundary point *s*_{2} instead of inside.

##### Remark 2.3.4.1

Note that in this case, a ball in the cross product was not a cross product of two balls, but the boundary of a ball in the cross product was cross product of the boundary of two balls. This leads us to:

##### Theorem 2.3.4.2:

*Let C and D be closeness spaces with underlying sets S and T, both of which consist of more than one point. Let space E = C ×, with underlying set U = S × T. Let ball B be a ball in E which does not contain any points (s _{2}, t_{2}) for any point s_{2} and some point t_{2}, contains all points (s_{3}, t_{3}) for any point t*

*Then B is not the cross product of two balls, but the boundary of B is the cross product of the boundaries of two balls. Furthermore, if the aforementioned boundray is nonempty and contains point u _{4} = (s_{4}, t_{4}), then the boundary consists of the cross product of the circle of radius (s_{1}, s_{4}) of radius (t_{1}, t_{4) centered at t1.}*

N.B. All of the hypotheses, which informally could be described as looking like clutter, are needed only to rule out degenerate exceptions. There are a number of equivalent replacements for the requirement that *B* contains no points at one *T* coordinate and all points at another.

**Proof:**

Proof by contradiction that *B* is not the cross product of two balls:

Assume that B is the cross product of two balls. *B* contains all of the points at one *T* coordinate, *t*_{3}, and none at another, *t*_{2}. Therefore, *B* contains *u*_{5} = (*s*_{5}, *t_{5} with s_{5} ≠ s_{1}. Every point (s_{1}, t_{6}) is closer to u_{1} than is u_{5}, so B contains a point at T coordinate t_{, and also does not contain that point. ⇒⇐}*

We consider two cases now:

###### Case 2.3.4.2.1:

*B* has an empty boundary. In that case, the further claim is vacuously true because the ‘if’ clause is not met. In addition, the former claim is also at least vacuously true: we observe that an entire space constitutes a ball, and the boundary of the entire space is empty. Therefore we examine the more interesting

###### Case 2.3.4.2.2:

*B* has a nonempty boundary. In that case, we observe that all points on the boundary are equidistant from some point *u*_{1}; if one were closer to another, we would have an exception to the triangle inequality.

I claim that if (*s*_{7}, *t*_{7}) and (*s*_{8}, *t*_{8}) are in the boundary *B*, then so are (*s*_{7}, *t*_{8}) and (*s*_{8}, *t*_{7}):

* Assume that (s_{7}, t_{7}) and (ss_{8}, t_{8} are in the boundary B. Then we can say the following, both for sets of points contained in B, and sets of points disjoint from B: there exists a set U_{7} ⊂ U containing points arbitrarily close to (s_{7}, t_{7}), and a set U_{8} ⊂ U containing points (s_{8}, t_{8}). U_{7} is a set of ordered pairs of points, which contain points of arbitrary close S coordinate to s_{7}, and a set U_{8} ⊂ U containing points arbitrarily close to (s_{8}*,

*t*

_{8}).

*U*

_{7}is a set of ordered pairs of points, which contains points of arbitrarily close

*S*coordinate to

*s*

_{7}, and arbitrarily close

*T*coordinate to

*t*

_{7}, and

*U*

_{8}is a set of ordered pairs of points, which contains points of arbitrarily close

*S*coordinate to

*s*

_{8}, and arbitrarily close

*T*coordinate to

*t*

_{8}. Take the cross product

*V*of the

*S*coordinates in

*U*

_{7}and the

*T*coordinates contained in

*U*

_{8}, and the cross product

*W*of the

*S*coordinates contained in

*U*

_{8}, and the

*T*coordinates contained in

*U*

_{7}. As this arguments applies both to sets of points contained in

*B*, and sets of points disjoint from

*B*, we have (

*s*

_{7},

*t*

_{8}) and (

*s*

_{8},

*t*

_{8}) as desired.

This establishes the independence of the *S* and *T* coordinates of points on the boundary, so the boundary is a cross product of some pair of sets in *S* and *T*.

This establishes the independence of the *S* and *T* coordinates of points on the boundary, so the boundary is a cross product of some pair of sets in *S* and *T*.

These sets must be equidistant from *s*_{1} and *t*_{1} respectively; if they were not, then we could select two radii of different length for the ball, and violate the triangle inequality. So they are subsets of the boundaries of balls; they must be the whole boundary because the cross product of two accumulation points of different sets is an accumulation point of the cross product of the two sets, as we argued above. And this establishes that the radius of the boundary must be the distance from the center to the cross product of two respective boundary points. So we have the boundary of *B*, for *u*_{4} = (*s*_{1}, *s*_{4}) a point on the boundary, equal to the crosss product of the circles of radius (*s*_{1}, *s*_{4}) and (*t*_{1}, *t*_{4}) centered at *s*_{1} and *t*_{1} respectively, as desired.

Q.E.D.

### Example 2.4:

Any subset of a closeness space is a closeness space.

### Remark 2.5:

The operations of taking a cross product of two closeness spaces in the dictionary order, and taking a subset of a closeness space, are together quite powerful. All other examples here are special cases of the operations taking a cross product of two closeness spaces in the dictionary order, and taking the subset of a closeness space.

#### Example 2.5.1:

The disjoint union of two closeness spaces *C* and *D*, in other words a union where *C* and *D* retain their closeness functions, and every function in one space is closer than every function in another space, is achievable by taking ℝ × *C*, paring it down until we have only a copy of (0, 1) where 0 is identified with a copy of *C*, and then taking the cross product of the result in *D*, and again paring it down until we only have a copy of (0, 1)where 0 is identified with a copy of *C* for which each element is identified with a single element, and 1 is identified with a copy of *D*.

If we allow not only finite but transfinite sequences of these two operations (which must be well-ordered, in order to be well-defined), then possible closeness spaces can take an almost unbelievable complexity beyond what is possible for metric spaces. The faintest hint of this is provided by a transfinite algorithm, and partial proof of correctness which is not reproduced here, which seems (given the Axiom of Choice) to be able to embed an arbitrary partial ordering in a totally ordered field. I believe that the power is sufficient to justify making:

#### Conjecture 2.5.2:

Assuming the Axiom of Choice, any closeness space can be generated from ℝ under the closeness arising from the usual Euclidean distance metric, by the operation of taking cross products in the dictionary order, and taking subsets.

### Example 2.6:

The long line appears to be a closeness space under what could intuitively be described as comparing the absolute value of differences. In general we cannot subtract ordinals as we can finite numbers, but we can do something comparable in this case.

We compare pairs of ordinals (*o*_{1}, *o*_{2}) and (*o*_{3}, *o*_{4}) as follows, in the case that both are distinct pairs:

Without loss of generality, assume that *o*_{1} < *o*_{2} and *o*_{3} < *o*_{4}.

*ƒ*(*o*_{1}, *o*_{2}, *o*_{3}, *o*_{4}) = ‘<‘ if *o*_{1} + *o*_{4} < *o*_{3} + *o*_{2}.

*ƒ*(*o*_{1}, *o*_{2}, *o*_{3}, *o*_{4}) = ‘=’ if *o*_{1} + *o*_{4} = *o*_{3} + *o*_{2}.

*ƒ*(*o*_{1}, *o*_{2}, *o*_{3}, *o*_{4}) = ‘>’ if *o*_{1} + *o*_{4} > *o*_{3} + *o*_{2}.

### Example 2.7:

The numbering of items in this thesis may be taken to be a finite and discrete closeness space, with closeness compared with a dictionary ordering on the numberings.

## Chapter 3: Towards Constructing a Field

We now define the next level of values:

### Definition 3.1:

A *level 2 value* is defined to be a level 1 sequences of values {*d*_{α}}_{α∈J} which is Cauchy convergent: for every element *ε*_{β} of level 1 epsilon {*e*_{β}}_{β∈K}, there exists an element *d*_{β} of {*d*_{α} such that every subsequent pair of values *d*_{α}1, *d*_{α}2 are within ε_{β} of each other.

What we are dong here is taking the closure of the set of level 1 values under the operation of taking limits, which might or might not be embeddable in ℝ and might be finer-grained. A level 1 value is included by a sequence that consists exclusively of that value.

### Note 3.2:

We compare two level 2 values *v*_{1} = {*d*_{1α}}_{α∈J} and *v*_{2} = {*d*_{2α}}_{α∈J} as follows:

If there is an element α_{0} of *J* such that, for all subsequent values of *α*_{1} and *α*_{2} we have α_{1} and α_{2} then *v*_{1} ≤ *v*_{2}.

If there is an element α_{0} of *J* such that, for all subsequent values of *α*_{1} and *α*_{2} we have α_{1} and α_{2} then *v*_{1} ≥ *v*_{2}.

If for every element α_{0} of *J*, there exist subsequent α_{1}, α_{2}, α_{3}, and α_{4} such that *d*_{1α1} ≤ *d*_{2α2} and *d*_{1α3} ≤ *d*_{2α4}, then *v*_{1} = *v*_{2.}

### Definition 3.3:

A *level 3 value* is defined to be an equivalence class of level 2 values under the partition induced by equality. Level 3 values are ordered in the same way that their members are ordered.

### Definition 3.4:

A *level 2 zero* is defined to be an infinite sequence of level 1 zeroe.

### Definition 3.5:

The *level 3 zero* is defined to be the equivalence class of the level 2 zeroes.

### Lemma 3.6:

The set of points whose distances are less than v_{1} from point s_{1}, for any value v_{1} and point s_{1}, constitutes a ball.

#### Proof:

It is clear that every point in this set is closer to *s*_{1} than is any point not in the set. So we need only to know that the set does not contain any boundary points.

If there is a boundary point *s*_{2}, then there is an epsilon at that boundary point contained in the set, and an epsilon at that boundary disjoint from the set. From these can be chosen a sequence of distances that converges to (*s*_{1}, *s*_{2}) and is inside the set, whereby *v*_{1} ≥ (*s*_{1}, *s*_{2}), and can also b chosen by a sequence of distances that converges to (*s*_{1}, *s*_{2}) and is outside the set, whereby *v*_{1} ≤ (*s*_{1}, *s*_{2}). So *v*_{1} = (*s*_{1}, *s*_{2}). The ball contains only points strictly closer than *v*_{1}, so it does not contain *s*_{2}. ⇒⇐

Q.E.D.

### Definition 3.7:

The *supremum* (resp. *infemum*) of a nonempty set *W* of level 2 values is defined to be the equivalence class containing the sequences *v*_{1} of values which satisfy the following three conditions:

#### Condition 3.7.1:

All elements of *v*_{1} are contained in some element of *W*.

#### Condition 3.7.2:

*v*_{1} contains at least one element greater than (resp. less than) or equal to any element of *W*.

#### Condition 3.7.3:

*v*_{1} is monotonically nondecreating (resp. nonincreasing).

#### Remark 3.7.4:

Not all sets will necessarily have a supremum or infemum. This a definition of what the supremum is if it exists, not necessarily a statement that one always exists.

There is at most a single equivalence class containing all such sequences, because any one contains an element greater than or equal to any element of any other, arbitrarily far along in the sequence.

The supremum and infemum of the empty set are undefined.

Now, we begin to develop an arithmetic.

### Definition 3.8:

A value *v*_{1} is said to be *equal to* *v*_{2} + *v*_{3} if *v*_{1} is the supremum over all triplets of points *s*_{1}, *s*_{2}, and *s*_{3} of the distance (*s*_{1}, *s _{3}), such that the following two conditions hold:*

#### Condition 3.8.1:

*( s_{1}, s_{2}0 ≤ v_{2}*

(s_{2}, s_{3v}

_{3}

#### Condition 3.8.2:

There do not exist any three points *s*_{4}, *s*_{5}, and *s*_{6} such that:

(*s*_{4}, *s*_{4}) ≤ *v*_{2}

(*s*_{5}, *s*_{6}) < *v*_{3}

(*s*_{4}, *s*_{6}) ≥ *v*_{1}

or

(*s*_{4}, *s*_{4}) < *v*_{2}

(*s*_{5}, *s*_{6}) ≤ *v*_{2}

(*s*_{4}, *s*_{6}) ≥ *v*_{1}

#### Notation 3.8.3:

A value *v*_{1} is said to be *a difference of v _{2} and v_{3}* if

*v*

_{2}=

*sub*

_{1}+

*v*

_{3.}

#### Remark 3.8.4

This definition does not guarantee the existence of a sum of two values; it only tells how to tell if a given value is equal to the sum of two others.

This value is chosen for its simplicity, specificity, and power; there numerous other ways of defining addition, some of which would seem to be a more generalized version of addition, doing to addition in ordered, cyclic, abelian groups as we know them what metric spaces do to ℝ^{2}. However, we will not investigate that generality here, and in particular, we are going to restrict our attention to a specific subset of closeness spaces, those for which addition as here defined is associative and uniquely defined.

If we not only do not restrict our attention, but replace the given condition with the stipulation that *v*_{1} is the supremum over points *s*_{1} of distances from *s*_{1} is the supremum over points *s*_{1} of distances from *s*_{1} to a point in the union of all closed balls of radius *v*_{3} whose centers lie in a closed ball of radius *v*^{2} centered at *s*_{1}, then we further lose commutativity; at least in the case of the long line, though, we have reproduced ordinal addition.

It appears that looking at those more general cases may be of mathematical interest and may allow the creation of an arithmetic that is looser and more general than that of an ordered, abelian group. However, we do not investigate that possibility here, and have not investigated it, beyond the brief attention paid in this remark.

This definition provides commutativity, and unique subtraction where defined (i.e. *v*_{1} – *v*_{2} may not be defined, but if it is, it is unique; provided that *v*_{1} – *v*_{2} may not be defined, but if it is, it is unique; provided that *v*_{1} = *v*_{2} + *v*_{3}, if *v*_{4} < *v*_{3}, then the distances between pairs of points eligible for the definition of addition will be less by at least a minimum positive amount, by the triangle inequality, so *v*_{2} + *v*_{4} < *v*_{2} + *v*_{3} = *v*_{1}. This observation, as well as establishing that subtraction is not ambiguous (though possibly undefined), proves for us:

#### Theorem 3.9:

For any three values *v*_{1}, *v*_{2}, and *v*_{3} for which *v*_{2} + is defined, we have:

If *v*_{1} < *v*_{2}, then *v*_{1} + *v*_{3} < *v*_{2} + *v*_{3}.

If *v*_{1} = *v*_{2}, then *v*_{1} + *v*_{3} = *v*_{2} + *v*_{3}.

If *v*_{1} > *v*_{2}, then *v*_{1} + *v*_{3} > *v*_{2} + *v*_{3}.

#### Sketch of Proof 3.9.1:

The first case is established above. The second case is established from the first case by the symmetry of an ordering, and the third case is established by the contradiction which would arise from the transitivity of the ordering if one sum was less than the other.

We now define our next level of value; as we earlier developed a closure under the operation of taking limits, we now define a closure under the operation of addition. Again, we are going with the more specific and powerful definition of addition given, at the loss of some generality.

### Hunch 3.10:

Addition of values is associative.

#### Suggestion of Proof Idea 3.10.1:

It seems that this arises from condition 3.8.2. Definition 3.8 is a refined version of earlier, less powerful definitions; we have not devoted enough time to the matter to establish associativity. We will continue on the assumption that this is true; one might say if need be that we are restricting our attention to spaces where addition is associative. We will further restrict attention to spaces which are closed under addition (although a slightly weaker condition is needed for my work, namely that any two level 4 values as defined below are uniquely comparable).

### Definition 3.11:

A *level 4 value* is defined to be a finite string of symbols as follows:

#### Part 3.11.1:

If *v*_{1} is a variable referring to a level 3 value, then “*v*_{1}” is a level 4 value.

#### Part 3.11.2:

If “*s*_{1}” and “*s*_{2}” are two level 4 values, then “(*s*_{1} + *s*_{2}) is a level 4 value.

#### Part 3.11.3:

If “*s*_{1}” is a level 4 value, then so is “— *s*_{1}“.

#### Part 3.11.4:

Nothing else is a level 4 value.

We compare level 4 values as follows:

#### Comparison 3.11.5:

If for level 3 values we have *v*_{1} < *v*_{2}, then for level 4 values, we have “*v*_{1}” < “*v*_{2}” and “— *v*_{1}” > ” — *v*_{2}“.

If for level 3 values we have *v*_{1} = *v*_{2}, then for level 4 values, we have “*v*_{1}” = “*v*_{2}” and “— *v*_{1}” > ” — *v*_{2}“.

If for level 3 values we have *v*_{1} > *v*_{2}, then for level 4 values, we have “*v*_{1}” > “*v*_{2}” and “— *v*_{1}” < ” — *v*_{2}“.

And we complete comparison by allowing certain manipulations, namely:

#### Part 3.11.6

If for level 3 values we have *v*_{1} = *v*_{2} + *v*_{3}, then inside a level 4 value *v*_{1} may be substituted or back-substituted for *v*_{2} + *v*_{3} and *v*_{3} + *v*_{2}, then inside a level 4 value *v*_{2} may be substituted or back-substituted for *v*_{1} + — *v*_{3}.

#### Part 3.11.7:

We may associate and commute while preserving equality.

#### Part 3.11.8:

We may add a like value to two different values without affecting their comparison.

#### Part 3.11.9:

Comparison is transitive.

We now define:

### Definition 3.12:

A *level 5 value* is an equivalence class of level 4 values under equality, with addition, additive inversion, and comparison of equivalence classes defined according to the equivalence classes of those operations on respective members.

We now have an ordered abelian group.

### Remark 3.13:

It is well known that an ordered abelian group may be embedded in a field. (Source: Anand Pillay).

### Definition 3.14:

For any two values *v*_{1} and *v*_{2}, *v*_{1} is said to be *of the same magnitude* as *v*_{2} if there exists a positive natural number *n* such that either *v*_{1} + *v*_{1} + ⋯ + *v*_{1} > *v*_{2} (with *v*_{1} added to itself *n*n times) and *v*_{2} + *v*_{2} + *v*_{2} ⋯ *v*_{2} >

*v*_{1}(with *v*_{2} added to itself *n* times), or *v*_{1} + *v*_{1} + ⋯ + *v*_{1} < — *v*_{2} (with *v*_{1} added to itself *n*n times) and — *v*_{2} — *v*_{2} — *v*_{2} ⋯ — *v*_{2} < *v*_{1} (with *v*_{2} added to itself *n* times).

It is clear that the magnitudes are equivalence classes of values.

The value 0 resides in its own magnitude, which will not be named.

#### Part 3.14.1:

Magnitude *M*_{1} is said to be *greater than* (resp. *less than*) magnitude *M*_{2} if it is a different magnitude, and *M*_{1} contains at least one positive value that is greater than (resp. less than) at least one positive value in *M*_{2.}

#### Part 3.14.2:

The magnitude which contains 1 is said to be *finite*.

All greater magnitudes than the finite magnitude are said to be *infinite*.

All lesser magnitudes than the finite magnitude (excluding the magnitude of 0), are said to be *infinitesimal*. The variable ε will hereafter refer to an infinitesimal.

To give a specific example of what kind of ordered field we have, let us look at

### Example 3.15

Let closeness space *C* be the space examined in example 2.3.4, namely ℝ^{2} × ℝ^{2}, under the closenesses induced by the dictionary order on Euclidean closenesses.

Then the closenesses are of type ℝ^{2} × ℝ^{2}, in the dictionary order.

The elements of a minimal imbedding field are of order type *S* ⊂ ℝ^{2}ℤ, such that all but finitely many of the coordinates of an element of *S* are zero.

Comparison of values is a dictionary comparison of their coordinates.

Addition of two values is coordinate-wise addition of reals. I.e. if *v*_{1} and *v*_{2} are values and *v*_{1i}, *v*_{2i} are the i^{th} coordinates of *v*_{1} and *v*_{2} respectively, then the i^{th} coordinate of *v*_{3i} of *v*_{3} = *v*_{1} + *v*_{2} is equal to *v*_{1i} + *v*_{2i}.

Multiplication of two values is as follows:

If *v*_{1} and *v*_{1} are as above, then *v*_{3} has coordinate *v*_{3i} equal to _{Σj+k=i} = *v*_{1j} × *v*_{2k.}

This is isomorphic to the field of ratios of polynomials in a single variable, over the real numbers. Note that, although the order type is fixed, a constant *c* chosen so that c^{0} = 1, c^{1} is a number of the lowest infinite magnitude and zero coordinate in other magnitudes, and *c*^{-1} is a number of the highest infinitesimal magnitudes, is not a unique constant. Any one such value can be arrived at by multiplying another such value by a nonzero real number.

The interpretation of this representation as given is that the 0-coordinate is the finite component, components of positive ℤ value are infinite components, and components of negative ℤ value are infinitesimal components (or vice versa).

Under this interpretation, we can say that the given closeness space is like a metric space, using the given field instead of ℝ as the measure. It could be stated to use the 0 coordinate for the large plane, and the — 1 coordinate for the miniature planes at each point of the large plan (in which case the space is interpreted as a roughly Euclidean plane with infinitesimal distances, or to use the 1-coordinate for the large plane, and the 0 coordinate for the small planes (in which case the space is interpreted as an infinite plane of real planes — it is to the Euclidean plane roughly as ω^{2} is to ω among ordinals), or indeed z and z – 1 for any integer z.

#### Example 3.15.1: Non-Standard Analysis

This allows achievement of at least some of the results of nonstandard analysis. For example:

##### Definition 3.15.1.1:

For the duration of this example, we define the *nearest real number* to a finite value to be the value of the same first coordinate, and zero component in the second coordinate. (I.e. a distance of 3.7 × 0 is the nearest real number to 3.7 × — 23.4. 3.7 × — 23.4 or 0 × 14 are not the nearest real numbers to anything.)

##### Definition 3.15.1.2:

We define the *limit of a function ƒ at point x* to be equal to the nearest real number to *ƒ*(*x* + ε), if such a real number exists and is uniquely defined across all infinitesimals ε.

For example, the limit of *ƒ*(*x*) = *x* + 1 at *x* = 1 is the nearest real number to

*ƒ*(*x* + ε) =

*f*(1 + ε) =

1 + ε + 1 =

2 + ε

which has 2 as its nearest real number.

##### Definition 3.15.1.3:

We define a function *ƒ*(*x*) to be *continuous at point x* if *f* is defined at *x* and if, for every infinitesimal *e*, we have ||*ƒ*(*x* + ε) — *ƒ*(*x*)|| at most an infinitesimal.

For example, if we have

*ƒ*(*x*) = *x* + 1 if *x* ≥ 0

*ƒ( x) = 0 otherwise*

then we have f continuous at -1 and 1, but not continuous at 0:

*ƒ*(-1 + ε) – *ƒ*(-1) = 0 – 0*ƒ*(1 + ε) – *ƒ(1) = 1 + ε + 1 – 1 + 1 = ε
but problems when we examine a negative value of ε with nearest real number at 0:*

If ε < 0, then *ƒ*(0 + ε) = *ƒ*(ε), but we have *f*(0) = 0 + 1 = 1, and 0 — 1 = &mdash 1 is not an infinitesimal.

##### Definition 3.15.1.4:

We define the derivative of a function *ƒ* at point *x* to be the nearest real number to

(*ƒ*(*x* + ε) – *ƒ*(*x*)) / ε

if such a number exists and is well-defined across all infinitesimals ε.

For example, the derivative of *ƒ*(*x*) = *x*^{2} is equal to the nearest real number to:

* x + ε)^{2} – x^{2}) / ε =*) / ε =

(x^{2} + 2xε + ε^{2} – x^{2}) / ε =

(2xε + &epsilon^{2}

2

*x*+ ε

and the nearest real number to 2*x* + ε is 2*x*. So we have the derivative of *x*^{2} equal to 2*x*.

## Closing remark

Providing a nonstandard analysis with derivatives seems straightforward enough; notwithstanding the fundamental theorem of calculus, it is not clear to this author how to adapt these findings to create an integral, although just as epsilon-delta arguments provide a finite workaround to infinitesimals, the core concept of integration in calculus find a finite workaround to summation of an infinite number of infinitesimally thick slices. It might be noted that this system does yet have the infinite sums and infinite integers of non-standard analysis. Perhaps our restricted attention disregarded some closeness spaces or other matters yielding fields that would allow a more powerful non-standard analysis; perhaps work with the closeness spaces involving the ordinals cross [0, 1) — a nonnegative long real number line — would achieve such things. However, we will draw a limit to the investigation here.