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.
The real number p is a limit point of a set X of real numbers provided that for every positive number , there is an element x of the set X such that .
As an example, let X consists of all real numbers of the two forms and , 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.
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.