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Topological spaces

Def. Topology (on a set). A topology on a set X is a collection τ of subsets of X, satisfying the following axioms:

(1) The empty set and X are in τ

(2) The union of any collection of sets in τ is also in τ

(3) The intersection of any finite number of sets in τ is also in τ

The elements of X are usually called points, although they can be any mathematical objects. The sets in τ are called open sets and their complements in X are called closed sets. Subsets of X may be either closed or open, neither closed nor open, or both closed and open. A set that is both closed and open is called a clopen set. The sets X and ∅ are both open and closed.

Def. Topological space. A topological space is a set X for which a topology τ has been specified.

Note. In defining a topological space some authors state, “a topological space is the pair (X, τ).” I ask this. If I say, “John is a man with black hair”, is that equivalent to saying, “John is the pair (man, black hair)”? Why do mathematicians like to do things like this? Are they trying to aid the person studying the abstract, abstruse subject of point-set topology for the first time or to thwart, confound and stymie him? To denote a topological space X with a topology τ by the notation (X, τ) is different than declaring that a topological space is the pair (X, τ).

One day many years ago I wanted to learn what the subject of topology was all about and picked up a textbook on it. I found the above definition on the first page. I simply stared at it with glazed eyes, pondered it a bit, and decided to come back again at some later date. Some time later I picked up another book on topology with the idea of giving it another try, read the same definition again, and put the book back on the bookshelf. That is the effect it had on me. I guess that is all I need say.

What is the origin of this definition? I think it would be safe to say that it is simply a listing of properties of the open point sets of one, two, and three dimensional space i.e. open intervals on the real line, open circles and rectangles in the plane and open spheres and open rectangular parallelepipeds of three dimensional space, etc. Then, the urge to generalize led to the current definition in which the terms “open sets” and “points” are simply terms with no particular meaning, carry-overs from the original concept relating to open point sets (but not to be confused with them).

Suppose Jack tells me he is going to define the term “horse” and then gives me the following definition:

Def. Horse. A horse is anything possessing the following properties:

1) Has organs for seeing

2) Has organs for hearing

3) Has an organ for eating

4) Has a means for locomotion

I tell him, “That definition is ridiculous! It doesn’t define a horse! By that definition tigers, rats, people, roaches, fish, birds, snakes and flies are all horses!” He says, “It is not ridiculous! That is what I choose to call a horse! I can define a horse in any way I wish!” I say, “What good is the definition? It includes everything.” He says, “It does not include everything. It does not include rocks, air, water, plants, thoughts, feelings and many other things.”

The field of topology would probably have remained a total mystery to me had I not later read the Russian series, “Mathematics, Its Content, Methods and Meaning”, Volumes I, II and III, which has a chapter on topology. This series is a really excellent series, head and shoulders above anything else I have read for mathematical exposition. I very highly recommend it. It is a very intelligent and able presentation of the basic concepts of mathematics, revealing a depth of understanding and insight far in excess of anything else I have seen. After reading the chapter on topology I was of the opinion that topology was a very interesting field. However, my feelings about the definition above are as negative as ever. The types of point sets that can qualify as a topological space under the above definition can be discrete or continuous. The definition tells me nothing, gives me no feel or concept of what kind of animal a topological space is. So what is a topological space? To answer that I will have to go to the Russian series and quote from it:

Topological space. “A topological space is any collection of points (an arbitrary collection of elements) in which a relation of neighborhood of one point to a set of points is defined and, consequently, a relation of neighborhood or adherence of two sets (figures) to one another. This is a generalization of the intuitive intelligible relation of neighborhood or adherence of figures in the ordinary space. ... Lobacevskii with remarkable foresight pointed out that of all the relations of figures, the most fundamental is the relation of neighborhood. .... As the subsequent development of topology has shown, it is precisely the property of neighborhood that underlies all other topological properties. .... The concept of neighborhood expresses the notion of a point being infinitely near to a set. Therefore every collection of objects in which there is a natural concept of continuity, of being infinitely near, turns out to be a topological space. .... The concept of a topological space is extremely general and the study of such spaces, abstract topology, represents the most general mathematical study of continuity. ..... Topology is the study of those properties of spaces, of figures in them, and of their transformations that are defined by the relation of neighborhood. The generality and fundamental nature of this relation makes topology into a very general geometrical theory that penetrates the diverse branches of mathematics, wherever continuity only is under discussion.

“The concepts of adjacency, neighborhood, infinite proximity .... are in essence the fundamental, primordial concepts of topology in the full extent in which we now understand this discipline.

“Every transformation of a geometric figure in which the relations of adjacency of various parts of the figure are not destroyed is called continuous: if the adjacencies are not destroyed, but also no new ones arise, then the transformation is called topological. Therefore, under a topological transformation of an arbitrary figure the parts of this figure that are in contact remain in contact, and the parts that are not in contact cannot come into contact; to put it briefly, in a topological transformation neither breaks nor fusions arise. In particular, two distinct points cannot be united into a single point. Therefore a topological transformation of any geometric figure, considered as the set of points forming it, is not only continuous, but also a one-to-one transformation: Any two distinct points of the figure are transformed into two distinct points. Thus topological transformations are single-valued and continuous both ways.

“Intuitively a topological transformation of an arbitrary geometric figure (a curve, a surface, etc.) can be represented in the following way. Let us imagine that our figure is made of some flexible and stretchable material, for example of rubber. Then it can be subjected to all possible continuous deformations under which it will be extended in some of its parts and contracted in others, and altogether will change its size and shape in every way.

“Suppose we imagine a spherical surface consisting of a thin rubber sheet. It can be stretched into many shapes but it cannot be changed into a square of a torus. A sphere and a torus are topologically distinct surfaces belonging to distinct topological types. We say that they are not homeomorphic to each other. A sphere and an ellipsoid, on the other hand belong, to the same topological type. They are homeomorphic to each other. This means that they can be carried into one another by a topological transformation.

“Every property of a geometrical figure that is preserved under an arbitrary topological transformation of it is called a topological property. Topology studies topological properties of figures. It also studies topological transformations and arbitrary continuous transformations of geometric figures.

Mathematics, Its Content, Methods and Meaning, Vol. III

Thus, from this, a topological space is a set of points that form a continuum, a set which has associated with it properties of continuity, adjacency and neighborhood.

The Funk & Wagnalls Dictionary gives the following definition of the word “topology”.

Def. Topology. The study of those properties of geometric figures or solid bodies that remain invariant under certain transformations.

The term “topological” suggests the idea of the properties of curves and surfaces thus implicitly implies concepts involving a continuum of points. The term “space” generally carries with it the idea of a continuum of points. Thus it would be natural to assume from the name “topological space” that a topological space would be concerned with point set continua. One could be excused for being confused by learning that this is not the case. That although some topological spaces (as the metric spaces) do represent point set continua, others are discrete sets of points, points completely isolated from one another, absent any continua. It seems odd and surprising that both types of point sets would be included under one name called a “topological space”. However, when one starts defining horses as we have above, that is what you get. One never knows what odd and strange creatures may qualify as a horse. That is the kind of thing you get when you use axiom-oriented definitions.

In a topological space an abstract structure has been defined in which the underlying points need not form a continuum and in which there is no notion of distance between points. In such a case, the ideas of neighborhood, interior point, exterior point, boundary point, limit point, continuous function, etc. as defined for metric spaces make no sense. Yet these same terms are used, with altered definitions, for a topological space. If we speak of a continuous function on a space of points that don’t form a continuum and where there is no concept of distance, what is the meaning of that?

What motivated the concept of a topological space? Why was this very abstract mathematical structure created? Presumably, there was a desire to create a mathematical structure that didn’t include the concept of distance in order to more effectively study point sets that didn’t have a natural concept of distance associated with them.

What is a topological space? What general features or properties can you state for it? It is like the horse defined above. It is hard to say what it is, hard to make any general statements about it. The point sets in some topological spaces, for example, represent points of a continuum. The point sets in other topological spaces (e.g. a discrete space) do not form a continuum. Since most spaces (for example, metric spaces or vector spaces) are collections of points from a continuum the term “space” may seem misleading. But then so are such terms as interior points, exterior points, limit points, continuous functions, etc. since they have the usual meanings only for some topological spaces (e.g. metric spaces) but make no sense for others.

When one abstracts from some concrete model to some very generalized, abstract concept and carries the terms of the concrete model over to the abstract structure he creates this kind of problem.

When one is talking about a metric space the terms interior point, exterior point, continuous function have their usual meaning and make sense. For a discrete space they don’t. Is the situation here like a computer algorithm that yields good numbers if the input is good, garbage if the input is bad, but it always yields numbers whether the input be good or bad? We can compute interior, exterior and limit points but whether they make sense or not is something else.

Topology is concerned with the properties of curves, surfaces and solids. Curves, surfaces and solids consist of point sets in the form of a continuum.

Examples of continua

One-dimensional continuum. A curve in 3-space. A curve in the plane. A curve in n-space.

Two-dimensional continuum. A surface in 3-space. The interior of a figure in 2-space e.g. interior of a square. A two-dimensional surface in n-space.

Three-dimensional continuum. A solid in 3-space. A three-dimensional surface in n-space.

What are the properties of one, two, three and n dimensional continua?

The model for the concept of a topological space is a continuum of one, two, three or more dimensions.

The definition of a topological space has been generalized from the model to include much more than the model. It is an axiomatic one which has been phrased to include not only the model but other things very different than the model. It includes things that have no property of adjacency, neighborhood or infinite proximity.

Open sets in space. Let S be the set of points comprising a closed surface such as a sphere or torus. Then every point in S is an interior point. Thus set S is an open set.

Let S be the set of points comprising a closed curve in two or three dimensional space.. Then every point in S is an interior point. Thus set S is an open set.

Let S be the set of points enclosed by a closed surface (but not including the surface) in three dimensional space. Then every point in S is an interior point. Thus set S is an open set.

Example: Interior of a sphere.

Let S be the set of points inside a patch (but not including the boundary) on a surface in 3-space. Then every point in S is an interior point. Thus set S is an open set.

If S is any open set in one, two, three or n dimensional space it represents a continuum of points in which the collection τ of all open subsets of S have the properties named in the definition of a topological space i.e.

1) The empty set and S are in τ

2) The union of any collection of sets in τ is also in τ

3) The intersection of any finite collection of sets in τ is also in τ

and thus qualifies as a topological space. It is precisely this kind of topological space, in which points form a part of a continuum, that the Russian series is referring to.

As we have noted a topological space as it has been defined does not include a “distance” concept — a distance concept is not a requirement for the space. That is, a set of points which meets only the minimum conditions to qualify as a topological space has no distance defined between points. Thus at its minimum, with no distance defined, a topological space is essentially a discrete space. Because there need be no distance defined the concepts of closeness, neighborhood, vicinity, continuity, etc. have no meaning in a topological space (as we understand the meanings from two and three dimensional space). Because of this, concepts such as interior point, exterior point, limit point, boundary point, open set, closed set defined for point sets in two and three dimensional space, have no natural meaning because their natural meanings in two and three dimensional space depend on the distance concept. A very basic space has been defined called a “topological space”which, if you add more properties to it, can become a space such as a metric space. But what are the properties and characteristics of that very basic space in its most general form? Should it be called topological if it is basically discrete? It is only made topological by adding conditions or properties to it.

Let us now ask the following question.

Q. Why would a very abstract mathematical structure be defined for the study of topological properties that encompassed both point-sets in continua and discrete point sets?

A. Discrete points sets need to be included because discrete point sets do arise in the investigation of topological problems in the form of isolated singular points. If the theory is to handle general point sets in n-space, isolated points must be included. Also, there is a close connection between topological ideas and the theory of infinite sequences. There is a lot of interplay between the two areas of investigation. And infinite sequences are discrete point sets. One needs a theoretical framework that can handle both kinds of point sets.

Perhaps a topological space should be called a pre-topological space that becomes topological when certain conditions are added.