On general topology and the relation of the properties of the class of all continuous functions to the properties of space

Abstract

aggregate or set P and a relation E'KE between subsets E, E' of P. The set E' is assumed to be unique and determined for each subset E of the fundamental set P. Thus the relation E' = K(E) defines a single-valued set-valued set-function, whose range is the class of all subsets of P, and whose values are also subsets of P. Different set-functions K(E), H(E), relative to the same set P determine different spaces. 2. Properties of sets of points. The elements of the class P in a space (P, K) will be called points. The elements, if any, of the set E'=K(E) will be called points of accumulation of the set E with reference to the function K(E), relation K, or space (P, K). The terms of the theory of sets of points retain their usual significance in relation to the points of accumulation defined by a set-function. In case several set-functions are to be considered simultaneously we indicate the function of reference by writing K-point, K-closed, K-interior, etc. 3. Derived set-functions. In this and following paragraphs we consider a number of set-functions which may be defined in terms of a given set function K(E), together with the corresponding spaces and types of points of accumulation. The complement P E of a set E may be represented by the set function C(E). In this notation the definition of interiority may be stated as follows. A point p is K-interior to a set E if it is not a K-point of any subset of C(E).* The set I(E) of all the K-interior points of a set E is of great importance. The function I(E) is easily seen to be monotonic increasing. That is, if E is a subset of a set G, I(E) 'I(G). Open sets are defined by the equation E = I(E), that is, they consist entirely of interior points. It is quite easy to show that the sum of a finite or infinite number of open sets is open. The corresponding proposition regarding the product of closed sets cannot be extended to general space. The frontier of a set E, F(E), is given by the formula F(E) = EK [C(E) ] + C(E) * K(E). The isolated points of a set E define the function S(E). 4. Iteration of set-functions. By finite iteration we obtain from a given set-function K(E) the derived sets and derived functions of finite order