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AN INTRODUCTION TO ALGEBRAIC TOPOLOGY
By Andrew H. Wallace Dover Publications, Inc.
Copyright © 2014 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-15295-0
CHAPTER 1
INTRODUCTION
1. Continuity and neighbourhoods
In analysis one's first introduction to the idea of continuity is usually based on the idea that a continuous function f of the real variable x should be such that small changes in x result in small changes in f(x); the requirement being in fact that the graph y = f(x) should not have any breaks in it, but should be, in the intuitive sense, a continuous curve. The next step in analysis is to make this notion precise by introducing the ε-terminology.
For the present purpose, however, it is more convenient to preserve a geometrical outlook, by noting first that the function f may be thought of as a mapping of the x-axis into the y-axis, each value of x being mapped on a uniquely determined value f(x) of y. The continuity of f at x' can then be expressed by saying that points sufficiently near to x' on the x-axis are mapped by f into points arbitrarily near f(x') on the y-axis.
In analysis the phrases "sufficiently near" and "arbitrarily near" would be expressed explicitly by means of inequalities. But the advantage of the geometrical language is that the same words can be used to define continuity in a more general situation, Namely, let Em and En be any two Euclidean spaces, of dimensions m and n respectively, and let f denote a mapping of some subset A of Em into En. That is to say, f is a law which assigns to each point of A a uniquely defined point of En. Then, as in the case of the function of one variable, f will be said to be continuous at the point p in A if all points of A sufficiently near to p are mapped into points arbitrarily near f(p) in En. It should be noted at this point that if n = 1 the notion just defined coincides with the analytical concept of a real valued function of m variables continuous in these variables. It should also be verified as an exercise that the mapping f of a subset of Em into En is continuous at a point with coordinates (x'1, x'2, ..., x'm) if and only if the coordinates (y1, y2, ..., yn) of the image under f of a variable point (x1, x2 ..., xm) of Em are continuous functions of x1, x2 ..., xm at (x'1, x'2, ..., x'm).
The concept of continuity of a mapping of one Euclidean space into another extends the notion of a continuous function of one variable, and has been phrased in such a way that the same words describe both situations. That it is desirable to try to extend this idea still further to mappings between sets other than subsets of Euclidean spaces may be seen by considering the following simple example. In analysis one often wants to work with a family of functions ft depending continuously on a parameter t. What one usually does is to think of the symbol ft(x) as if it were a function of two variables f(t, x) continuous or uniformly continuous in the first. But, logically, this means a shift of view-point, since each ft is a function defined on some set of real numbers, while f, defined so that f(t, x) = ft(x), is an operator defined on some set in the (t, x)-plane. It would be more satisfactory if one could describe what one meant by a continuous family of functions without this change of terminology.
A convenient and natural way of reformulating the notion of a continuous family of functions is to think first of dependence on a parameter t as being given by a mapping t -> ft of the real numbers into the set of all real valued functions of a real variable. Then to say that the ft depend continuously on t one would like to say that this mapping is continuous. In order that one may be able to do this it is necessary to know what one means by saying that two functions ft and ft are near to one another; if this idea is defined then the same wording can be used to define the continuity of the mapping t -> ft has been used already to define the continuity of a mapping between Euclidean spaces.
The remark made in the last sentence, although concerned with a special example, is much more general in implication. Namely, if A and B are any sets of objects of any kind among which a concept of nearness is defined, then the idea of a continuous mapping of A into B can be formulated. The example of a continuous family of functions shows that such a general notion of continuity has applications: many more examples could be given, and indeed will be in later sections, to show that the idea is worth following up.
2. The abstract concept of neighbourhood
The idea introduced at the end of the last section, namely of a set A along with a definition of nearness of two elements of A, is essentially the starting point of the subject of point-set topology. The idea is not quite in its most satisfactory form, since nearness in its only familiar form so far, that is in Euclidean space, is measured by a distance formula; and it is not always convenient, or even possible, to give such a numerical measure in more general cases. That this defect can be remedied will now be seen from a further analysis of the idea of continuity of a mapping f of a set in a Euclidean space into another Euclidean space.
If one adopts the rather natural course of calling points of a Euclidean space near a point p a neighbourhood of p, then to say that a mapping f of a subset A of Em into En is continuous at p means that, if U is a preassigned arbitrarily small neighbourhood of f(p), then all the points of a sufficiently small neighbourhood V of p are mapped into U. This is simply a restatement of the definition of continuity already given in §1. The first thing to notice about this restatement is that one can actually omit the word "small," and simply say that f is continuous at p if, for a preassigned neighbourhood U of f(p), there exists a neighbourhood V of p all of whose points are mapped into U by f. Naturally the preassigned neighbourhoods of f(p) will include arbitrarily small ones, and common sense shows that the corresponding V must be sufficiently small; but there is no need to say so.
The omission of the word "small" may seem a trivial matter, but it is a logical step forward. For the restated definition of continuity at p does not contain any explicit mention of the idea of distance, but is formulated in terms of neighbourhoods of p and f(p). That is to say, continuity at p is defined in terms of certain families of point-sets assigned to the points p and f(p), and called neighbourhoods of p and f(p) respectively. Of course, in defining these neighbourhoods the idea of distance has to be used. But a separation of the continuity into two stages has been effected, as stated in the following definitions:
Definition A. A neighbourhood of a point p in a Euclidean space is the set of all points within a distance r of p for some r.
Definition B. If f is a mapping of a set A in a Euclidean space into another Euclidean space, then f is continuous at a point p if, given any neighbourhood U of f(p), there is a neighbourhood V of p such that f(V [intersection] A) [subset] U.
The significance of this separation of ideas is that it may be possible to assign to each member p of an abstract set A (not necessarily now a subset of a Euclidean space) a family of subsets of A to be called neighbourhoods of p, a similar assignment being made to the elements of a second abstract set B. This having been done, one can define a mapping of A into B to be continuous at p if, for any preassigned neighbourhood U of f(p), there exists a neighbourhood V of p such that all members of V are mapped into U by f. Of course this assignment of neighbourhoods to the members of an abstract set cannot be made in an arbitrary way. For example, one obviously wants a set which is to be called a neighbourhood of p actually to contain p. To see what further conditions must be imposed on a family of subsets of an abstract set A before they can reasonably be called neighbourhoods of some member of A, a closer inspection will be made of the situation in a Euclidean space.
As already pointed out, the concept of continuity can be adequately defined if one takes as neighbourhoods of a point p in Euclidean space the family of all spheres with centre p. But as this definition of neighbourhoods is rather too closely tied up with the notion of distance, the following question will be considered first: what is the most general meaning one can give to the word "neighbourhood" without changing the meaning of continuity? To make this question more explicit, consider real-valued functions defined on a Euclidean space. And suppose that, according to some law or other, each point p of this space has assigned to it a family of subsets called N(p). Then the following property of a real-valued function f will be called the property C:
The function f will be said to have the property C at a point p if, for any preassigned positive number ε, there exists a set U belonging to N(p) such that |f(p') - f(p)| < ε for all p' in U. In any case p [member of] U for all U in N(p) will be assumed.
Then the question being asked is: what is the most general definition which can be made for the families N(p) such that, for every function f,f has the property G at a point p if and only if it is continuous (in the ordinary sense) at p?
(Continues...)
Excerpted from AN INTRODUCTION TO ALGEBRAIC TOPOLOGY by Andrew H. Wallace. Copyright © 2014 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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