Mathematical Morsel III: on sets and functions

Posted on by Nalin Pithwa

Reference: Chapter 1, Sets and Functions; Topology and Modern Analysis, G F Simmons, Tata McGraw Hill Pub.

It is sometimes said that math is the study of sets and functions. Naturally, this oversimplifies matters, but it does come as close to the truth as an aphorism can.

The study of sets and functions leads two ways. One path goes down, into the abysses of logic, philosophy, and the foundations of mathematics. The other goes up into the highlands of mathematics itself where these concepts are indispensable in almost all of pure mathematics as it is today. Needless to say, we follow the latter course. We regard sets and functions as tools of thought, and our purpose in this chapter is to develop these tools to the point where they are sufficiently powerful to serve our needs through the rest of the book.

As the reader proceeds, he will come to understand that the words set and function are not as simple as they may seem. In a sense, they are simple, but they are potent words, and the quality of simplicity they possess is that which lies on the far side of complexity. They are like seeds, which are primitive in appearance but have the capacity for vast and intricate developments.

Mathematical Morsel I: How to read math books

Posted on by Nalin Pithwa

A note to reader of Math books:

(I like this v much; I am reproducing it verbatim as the advice of G F Simmons, Topology and Modern Analysis, Tata McGraw Hill Publication, India)

Two matters call for special comment: the problems and proofs.

The majority of this problems are corollaries and extensions of theorems proved in the text, and are freely drawn upon at all later stages of the book. In general, they serve as a bridge between ideas just treated and development yet to come, and the reader is strongly urged to master them as he goes along.

In the earlier chapters, proofs are given in considerable detail, in an effort to smooth the way for the beginner. As our subject unfolds through the successive chapters and the reader acquires experience in following abstract mathematical arguments, the proofs become briefer and minor details are more and more left for the reader to fill in for himself. The serious student will train himself to look for gaps in proofs, and should regard them as tacit invitations to do a little thinking on his own. Phrases like “it is easy to see,” “one can easily show,” “evidently,” “clearly,” and so on are always to be taken as warning signals which indicate the presence of gaps, and they should put the reader on his guard.

It is a basic principle in the study of mathematics, and one too seldom emphasised, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it as a single idea. In achieving this and, much more is necessary than merely following the individual steps in the reasoning. This is only the beginning. A proof should be chewed, swallowed, and digested, and this process of assimilation should not be abandoned until it yields a full comprehension of the overall pattern of thought.

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Isn’t that beautiful advice for all budding math aspirants ?

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Cheers,

Nalin Pithwa.

Problem Set based on VII. Complete Metric Spaces

Posted on by Nalin Pithwa

Problem 1.

Prove that the limit f(t) of a uniformly convergent sequence of functions \{ f_{n}(t)\} continuous on [a,b] is itself a function continuous on [a,b].

Hint: Clearly,

|f(t)-f(t_{0})| \leq |f(t) - f_{n}(t)|+|f_{n}(t) - f_{n}(t_{0})|+|f_{n}(t_{0}) - f(t_{0})| where t, t_{0} \in [a,b].

Use the uniform convergence to make the sum of the first and third terms on the right small for sufficiently large n. Then, use the continuity of f_{n}(t) to make the second term small for t sufficiently close to t_{0}.

Problem 2.

Prove that if R is complete, then the intersection \bigcap_{n=1}^{\infty}S_{n} figuring in the theorem 2 consists of a single point. Theorem 2 is recalled here: Nested sphere theorem: A metric space R is complete if and only if nested sequence S_{n} = \{ S[x_{n}, r_{n}]\} of closed spheres in R such that t_{n} \rightarrow 0 as n \rightarrow \infty has a non empty intersection \bigcap_{n=1}^{\infty}S_{n}.

Problem 3.

Prove that the space m is complete. Recall: Consider the set of all bounded infinite sequences of real numbers x = (x_{1}, x_{2}, \ldots, x_{k}, \ldots) and let \rho(x,y) = \sup_{k}|x_{k}-y_{k}|. This is a metric space which we denote by m.

Problem 4:

By the diameter of a subset A of a metric space R is meant the number d(A) = \sup_{x, y \in A} \rho(x,y)

Suppose R is complete and let \{ A_{n}\} be a sequence of closed sets of R nested in the sense that

A_{1} \supset A_{2} \supset \ldots \supset A_{n} \supset \ldots

Suppose further that

\lim_{n \rightarrow \infty} d(A_{n})=0.

Prove that the intersection \bigcap_{n=1}^{\infty}A_{n} is non empty.

Problem 5.

A subset A of a metric space R is said to be bounded if its diameter d(A) is finite. Prove that the union of a finite number of bounded sets is bounded.

Problem 6.

Give an example of a complete metric space R and a nested sequence \{ A_{n}\} of closed subsets of R such that

\bigcap_{n=1}^{\infty}A_{n} = \phi.

Reconcile this example with Problem 4 here.

Problem 7.

Prove that a subspace of a complete metric space R is complete if and only if it is closed.

Problem 8.

Prove that the real line equipped with the distance metric function

\rho(x,y) = |\arctan{x} - \arctan{y}| is an incomplete metric space.

Problem 9.

Give an example of a complete metric space homeomorphic to an incomplete metric space.

Hint: Consider the following example we encountered earlier: The function y = f(x) = \frac{2}{\pi}\arctan{x} establishes a homeomorphism between the whole real line (-\infty, \infty) and the open interval (-1,1).

Comment: Thus, homeomorphic metric spaces can have different metric properties.

Problem 10:

Carry out the program discussed in the last sentence of the following example:

Ir R is the space of all rational numbers, then R^{*} is the space of all real numbers, both equipped with the distance \rho(x,y) = |x-y|. In this way, we can “construct the real number system.” However, there still remains the problem of suitably defining sums and products of real numbers and verifying that the usual axioms of arithmetic are satisfied.

Hint: If \{ x_{n}\} and \{ y_{n}\} are Cauchy sequences of rational numbers serving as “representatives” of real numbers x^{*} and y^{*} respectively, define x^{*} = y^{*} as the real number with representative \{ x_{n} + y_{n}\}.

I will post the solutions in about a week’s time.

Regards,

Nalin Pithwa.

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What is Topology about? Hocking and Young’s views

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Reference: Topology by Hocking and Young, Dover Publications Inc., NY.

Topology may be considered as an abstract study of the limit point concept. As such, it stems in part from a recognition of the fact that many important mathematical topics depend entirely upon the properties of limit points. The very definition of a continuous function is an example of this dependence. Another example is the meaning of the connectedness of a geometric figure. To exaggerate, one might view topology as the complement of modern algebra in that together they cover the two fundamental types of operations found in mathematics.

In applying the unifying principle of abstraction, we study concrete examples and try to isolate the basic properties upon which the interesting phenomena depend. In the final analysis, of course, the determination of the “correct” properties to be abstracted is largely an experimental process. For instance, although the limit of a sequence of real numbers is a widely used idea, experience has shown that a more basic concept if that of a limit point of a set of real numbers.

DEFINITION 1-1:

The real number p is a limit point of a set X of real numbers provided that for every positive number \epsilon, there is an element x of the set X such that 0 < |p-x|< \epsilon.

As an example, let X consists of all real numbers of the two forms \frac{1}{n} and \frac{n-1}{n}, where n is an integer greater than 2. Then, 0 and 1 are the only limit points of X. Thus, the limit point of a set need not belong to the set. On the other hand, every real number is a limit point of the set of all rational numbers, indicating that a set may have limit points belonging to itself.

Some terminology is needed before we pursue this abstraction further. Let S be any set of elements. These may be such mathematical entities as points in the Euclidean plane, curves in a given class, infinite sequences of real numbers, elements of an algebraic group, etc., but in general, we take S to be an abstract undefined set. To reflect the geometric content of topology, we refer to the elements of S by the generic name point. We may now name our fundamental structure.

DEFINITION 1-2:

The set S has  a topology (or is topologized) provided that, for every point p in S and every subset X of S, the question, “Is p a limit point of X?” can be answered.

This definition is extremely generally as to be almost useless in practice. There is nothing in it to impose certain desirable properties upon the limit point relation (to be discussed in detail later in the text), and also nothing in it indicates whereby the pertinent question can be answered. An economical method of accomplishing the latter is to adopt some rule or test whose application will answer the question in every case. For the set of real numbers, Definition 1-1 serves this purpose and hence, defines a topology for the real numbers. [The use of the word topology here differs from its use as the name of a subject. Loosely speaking, topology (the subject) is the study of topologies. (as in definition 1-2)].

A set S may be assigned different topologies but there are two extremes. For the first, we always answer the question in Definition 1-2 in the affirmative; that is, every point is a limit point of every subset. This yields a worthless topology: there are simply too many limit points!! For the other extreme, we assume that the answer is always “no,” that is, no point is a limit point of any set. The resulting topology is called the discrete topology for S. The very fact that it is dignified with a name would indicate that this extreme is not quite so useless as the first.

Those factors that dictate the choice of a topology for a given set S should become more apparent as we progress. In many cases, a “natural” topology exists, a topology agreeing with our intuitive idea of what a limit point should be. Definition 1-1 furnishes such a topology for the real numbers, for instance. In general, however, we require only a structure within the set S which will define limit point in a simple manner and in such a way that certain basic relations concerning limit points are maintained. To illustrate this latter requirement, it is intuitively evident that if p is a limit point of a subset X and X is contained in another subset Y, then we would want p to be also a limit point of Y. There are many such structures one may impose upon a set and we will develop the more commonly used topologies (in this chapter in the text). Before doing this, however, we continue our preliminary discussion with a few general remarks upon the aims and tools of topology.

The study of topologized sets (or any other abstract system) involves two broad and interrelated questions. The first of these concerns the investigation and classification of the various concrete realizations, or models, which we may encounter. This entails the recognition of equivalent model, as is done for isomorphic groups or congruent geometric figures, for example. In turn, this equivalence of models is usually defined in terms of a reversible transformation of one model onto another. This equivalence transformation is so chosen as to leave invariant the fundamental properties of the models. As examples, we have the rigid motions in geometry, the isomorphisms in group theory, etc.

One of the first to perceive the importance of these underlying transformations was Felix Klein. In his famous Erlanger Program (1870), he characterized the various geometries in terms of these basic transformations. For instance, we may define the Euclidean geometry as the study of those properties of geometric figures that are invariant under the group of rigid motions. (For example, Dihedral groups).

The second broad question in studying an abstract system such as our topologized sets involves consideration of transformations more general than the one-to-one equivalence transformations. The requirement that the transformation be one-to-one and reversible is dropped and we retain only the requirement that the basic structure is to be preserved. The homomorphisms in group theory illustrate this situation. In topology, the corresponding transformations are those that preserve limit points. Such a transformation is said to be continuous and is a true generalization of the continuous functions used in analysis. It follows that the second aspect of topology finds many applications in function theory.

Regards,

Nalin Pithwa.

What is Topology about? A nice accessible beginning viewpoint

Posted on by Nalin Pithwa

Reference: Intuitive Concepts in Elementary Topology by B H Arnold, Dover Publications, Inc. NY. 

What is Topology?

It is surprising that a fairly satisfactory description of topology can be obtained by changing “geometry” to “topology” , “geometric” to “topological” .

Let us back track a bit —- Euclidean geometry is the study of certain properties of figures in a plane or in space. Not all properties of a figure are of interest — only the geometric properties. For example, the colour of a triangle is not its geometric properties. But length of sides of a triangles, the measures of its angles, its area are certainly geometric properties. So, which properties are geometric ? Answer: Two figures are said to be congruent if and only if one of them can be placed upon the other so that the two figures exactly coincide.

In the above definition of congruence: the emphasis is on the phrase “can be placed upon.” Let us examine the phrase more closely. How do we “place” a figure? How can we move it? What are we allowed to do to it on the way? In geometry, the movements we are allowed are the rigid motions, (translations, rotations, and reflections) in which the distance between any two points of the figure is not changed. Thus, the geometric properties are those that are invariant under rigid motions.

In topology, the movements we are allowed might be called the elastic motions. We imagine that our figures are made of perfectly elastic rubber and in moving a figure, we can stretch, twist, pull and bend it at pleasure. We are even allowed to cut such a rubber figure and tie in a knot, provided that we later sew up the cut exactly as it was before; that is, so that points which were close together before we cut the figure are close together after the cut is sewed up. However, we must be careful that distinct points in a figure remain distinct; we are not allowed to force the two different points to coalesce into just one point. Two figures are said to be topologically equivalent if one figure can be made to coincide with the other by an elastic motion. The topological properties of a figure are those which are also enjoyed by all topologically equivalent figures. Thus, to a topologist, a coffee cup and a dough nut are the same ! 🙂

Certainly, any topological property of a figure is also a geometric property of that figure, but many geometric properties are not topological properties. The topological properties of a figure can be only the most basic and fundamental of its geometric properties. In topology, when we do elastic transformations, the properties of the figures based on “metric” are lost. That is, lengths, areas, measures of angles are not preserved. So what is preserved? Let us leave this question for some time and play with our elementary view of topology as rubber sheet geometry.

In fact, it might first appear at first glance that no property is a topological one — that, any property of a figure could be changed by an elastic motion ! 🙂 Fortunately, this is not the case. For instance, a circle C divides the points of a plane into 3 sets — the points inside the circle, the points on the circle and the points outside the circle. Try to visualize this…squeeze…smash the circle…whatever is in the interior will still be in the interior, whatever is on the circle will still be on the circle and whatever is outside it will remain outside it ! 🙂 So, this is a topological property of a circle ! 🙂

Regards,

Nalin Pithwa

 

What is Topology about? Paul Alexandroff’s view(s)

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Reference: Elementary Concept of Topology by Paul Alexandroff (Translated by Alan E. Farley). Dover Publications, Inc. NY.

Prof. David Hilbert’s views via the preface to this little booklet: (June 1962, Gottingen).

Few branches of geometry have developed so rapidly and successfully in recent times as topology, and rarely has an initially unpromising branch of a theory turned out to be of such fundamental importance for such a great range of completely different fields as topology. Indeed, today in nearly all branches of analysis and in its far reaching applications, topological methods are used and topological questions asked.

Such a wide range of applications naturally requires that the conceptual structure be of such precision that the common core of the supeficially different questions may be recognized. It is not surprising that such an analysis of fundamental geometrical concepts must rob them to a large extent of their immediate intuitiveness —- so much the more, when in the application to other fields, as in the geometry of our surrounding space, an extension to arbitrary dimensions becomes necessary.

While I have attempted in my Anschauliche Geometrie to consider spatial perception, here it will be shown how many of these concepts may be extended and sharpened and thus, how the foundation may be given for a new, self-contained theory of a much extended concept of space. Nevertheless, the fact that again and again vital intuition has been the driving force, even in the case of all of these theories, forms a glowing example of the harmony between intuition and thought.

Thus, the following book is to be greeted as a welcome complement to my Anschauliche Geometrie on the side of topological systematization; may it win new friends for the science of geometry.

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I. The specific attraction and in a large part the significance of topology lies in the fact that its most important questions and theorems have an immediate intuitive content and thus teach us in a direct way about space, which appears as the place in which continuous processes occur. As confirmation of this view, I would like to begin by adding a few examples to the many known ones:

i) One need only think of the simplest fixed-point theorem or of the well-known topological properties of closed surfaces such as are described, for instance, in Hilbert and Cohn-Vossen’s Anschauliche Geometrie, chapter 6. Published in English under the title Geometry and the Imagination.

2) The Jordan Curve Theorem:

A simple closed curve (that is, the topological image of a circle) lying in the plane divides the plane into precisely two regions and forms their common boundary.

3. The question which naturally arises now is: What can one say about a closed Jordan curve theorem in three-dimensional space?

The decomposition of the plane by this closed curve amounts to the fact that there are pairs of points which have the property that every polygonal path which connects them (or, which is “bounded” by them) necessarily has points in common with the curve. Such pairs of points are said to be separated by the curve or “linked” with it.

In three-dimensional space, there are certainly no pairs of points which are separated by our Jordan curve, but there are closed polygons which are linked with it in the natural sense that every piece of surface which is bounded by the polygon necessarily has points in common with the curve. Here the portion of the surface spanned by the polygon need not be simply connected but may be chosen entirely arbitrarily.

The Jordan curve theorem may also be generalized in another way for three-dimensional space : in space, there are not only closed curves, but also closed surfaces, and every such surface divides the space into two regions —- exactly as a closed curve did in the plane.

Supported by analogy, the reader can probably imagine what the relationships are in four-dimensional space: for every closed curve, there exists a closed surface linked with it; for every closed three-dimensional manifold a pair of points linked with it. These are special cases of the Alexander duality theorem.

((PS: In one dimension, a manifold may be a straight line, in two dimensions a plane, or the surface of a cube, a balloon, or a doughnut. The defining feature of a manifold is that, from the vantage point of any spot on such an object, the immediate vicinity looks like perfectly regular and normal Euclidean space. Think of yourself shrunk to the size of a pinpoint, sitting on the surface of a doughnut. Look around you, and it seems that you’re sitting on a flat disk. Go down one dimension and sit on a curve, and the stretch nearby looks like a straight line. Should you be perched on a three-dimensional manifold, however esoteric, your immediate neighborhood would look like the interior of a ball. In other words, how the object appears from afar may be quite different from the ,way it appears to your nearsighted eye.

By 1950, topologists were having a field day with manifolds, redefining every object in sight topologically. The diversity and sheer number of manifolds is such that today, although all two-dimensional objects have been defined topologically, not all three- and four-dimensional objects-of which there is literally an infinite assortment- have been so precisely described. Manifolds turn up in a wide variety of physical problems, including some in cosmology, where they are often very hard to cope with. The notoriously difficult three-body problem proposed by King Oskar II of Sweden and Norway in 1885 for a mathematical competition in which Poincar6 took part, which entails predicting the orbits of any three heavenly bodies–such as the sun, moon, and earth-is one in which manifolds figure largely.

This has been a Clay Millennium Problem and has been resolved by Grigory Perelman.))

4. Perhaps, the above examples leave the reader with the impression that in topology nothing at all but obvious things are proved !! 😦 

This impression will fade quickly as we go on. However, be that as it may, even these “obvious” things are to be taken much more seriously : one can easily give examples of propositions which sound as “obvious” as the Jordan curve theorem, but which may be proved false. Who would believe, for example, that in a plane there are three (four, firve….in fact, infinitely many!) simply connected bounded regions which all have the same boundary; or that one can find in three-dimensional space a simple Jordan arc (that is, a topological image of a polygonal line) such that there are circles outside of this arc that cannot possibly be contracted to a point without meeting it ? There are also closed surfaces of genus zero which possess an analogous property. In other words, one can construct a topological image of a sphere and an ordinary circle in its interior in such a way that the circle may not be contracted to a point wholly inside the surface.

5. All of these phenomena were wholly unsuspected at the beginning of the 20th century; the development of set theoretic methods  in topology first led to their discovery and; consequently, to a substantial extension of our idea of space. However, let me at once issue the emphatic warning that the most important problems of set theoretic topology are in no way confined to the exhibition of, so to speak, “pathological” geometrical structures; on the contrary, they are concerned with something quite positive. I would formulate the basic problem of set theoretic topology as follows:

To determine which set theoretic structures have a connection with the intuitively given material of elementary polyhedral topology and hence, deserve to be considered as geometrical figures —even if very general ones.

Obviously implicit in the formulation of this question is the problem of a systematic investigation of structures of the required type, particularly with reference to those of their properties which actually enable us to recognize the above mentioned connection and so bring about the geometrization of the most general set theoretic topological concepts.

PS: The purpose of sharing or blogging this is just to revise it for myself and hopefully, some readers will also benefit. In particular, these are my own study notes. Especially, Prof Paul Halmos used to say “I write what I talk to myself. I think by writing”…

Regards,

Nalin Pithwa.

Cauchy’s Mean Value Theorem and the Stronger Form of l’Hopital’s Rule

Posted on by Nalin Pithwa

Reference: G B Thomas, Calculus and Analytic Geometry, 9th Indian Edition. 

The stronger form of l’Hopital’s rule is as follows:

Suppose that f(x_{0})=g(x_{0})=0 and the functions f and g are both differentiable on an open interval (a,b) that contains the point x_{0}. Suppose also that g^{'} \neq 0 at every point in (a,b) except possibly x_{0}. Then,

\lim_{x \rightarrow x_{0}} \frac{f(x)}{g(x)} = \lim_{x \rightarrow x_{0}}\frac{f^{'}(x)}{g^{'}(x)}…call this I, provided the limit on the right exists.

Remarks:

The proof of the stronger from of l’Hopital’s rule in based on Cauchy’s mean value theorem, a mean value theorem that involves two functions instead of one. We prove Cauchy’s theorem first and then show how it leads to l’Hopital’s rule.

Cauchy’s Mean Value Theorem:

Suppose the functions f and g are continuous on [a,b] and differentiable through out (a,b) and suppose also that g^{'} \neq 0 through out (a,b). Then, there exists a number c in (a,b) at which

\frac{f^{'}(c)}{g^{'}(c)} = \frac{f(b)-f(a)}{g(b)-g(a)}.

(Note this becomes the ordinary mean value theorem when g(x)=x).

Proof of Cauchy’s Mean Value theorem:

We apply the ordinary mean value theorem twice. First, we use it to show that g(b) \neq g(a). Because if g(b)=g(a), then the ordinary Mean Value theorem says that

g^{'}(c) = \frac{g(b)-g(a0}{b-a}=0 for some c between a and b. This cannot happen because g^{'}(x) \neq 0 in (a,b).

We next apply the Mean Value Theorem to the function

F(x) = f(x)-f(a) - \frac{f(b)-f(a)}{g(b)-g(a)}(g(x)-g(a))

This function is continuous and differentiable where f and g are, and note that F(b)=F(a)=0. Therefore, by the ordinary mean value theorem, there is a number c between a and b for which F^{'}(c)=0. In terms of f and g, this says

F^{'}(c) = f^{'}(c) - \frac{f(b)-f(a)}{g(b)-g(a)}(g^{'}(c)) = 0

or \frac{f^{'}(c)}{g^{'}(c)} = \frac{f(b)-f(a)}{g(b)-g(a)} which is equation II above.

Proof of the stronger form of L’Hopital’s Rule:
We first establish equation I for the case \lim x \rightarrow x_{0}^{+}. The method needs almost no change to apply to the case \lim x \rightarrow x_{0}^{-}, and the combination of these two cases establishes the result.

Suppose that x lies to the right of x_{0}. Then, g^{'}(x) \neq 0 and we can apply Cauchy’s Mean Value theorem to the closed interval from x_{0} to x. This produces a number c between x and x_{0} such that

\frac{f^{'}(c)}{g^{'}(c)} = \frac{f(x)-f(x_{0})}{g(x)-g(x_{0})}

But, f(x_{0})=g(x_{0})=0 so

That \frac{f^{'}(c)}{g^{'}(c)}= \frac{f(x)}{g(x)}

As x approaches x_{0}, c approaches x_{0} as it lies between x and x_{0}. Therefore,

\lim_{x \rightarrow x_{0}^{+}} \frac{f(x)}{g(x)} = \lim_{x \rightarrow x_{0}^{+}}  \frac{f^{'}(c)}{g^{'}(c)} = \lim_{x \rightarrow x_{0}^{+}} \frac{f^{'}(x)}{g^{'}(x)}.

This establishes l’Hopital’s Rule for the case where approaches x_{0} from right. The case where x approaches x_{0} from the left is proved by applying Cauchy’s Mean Value Theorem to the closed interval [x,x_{0}] when x < x_{0}.

QED.

Regards,

Nalin Pithwa