On General Topology and the Relation of the Properties of the Class of All Continuous Functions to the Properties of Space
Transactions of the American Mathematical Society
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ON GENERAL TOPOLOGY 291 type of abstract set a topological space. For the same reason Fréchet* has proposed that the term topological space refer to the most general form of abstract set, the case in which the points of accumulation are defined by an arbitrary set-function. This terminology will be used throughout the following discussion. A topological space of the Hausdorff type will be
... dorff type will be referred to as a Hausdorff space. The first part of the present paper may be regarded as a discussion of the following problem: Determine the extent of the general theory of topological spaces, f It is also a study of various set-valued set-functions which may be defined in terms of a postulated arbitrary set-function K(E). By the aid of suitable definitions many important theorems of the theory of sets of points are extended to the most general topological space. Several of the set-functions defined by a given set-valued set-function K(E) are of particular interest. The function V(E) of §9 is associated with the concept neighborhood and defines a neighborhood space associated with the given topological space. The class of proper nuclear points} of a set E defines a set-function discussed in §11. The importance of this set-function was first recognized by C. Kuratowski and W. Sierpinski § who used it in the solution of the following problem of Fréchet: Determine the most general class (L) in which the theorem of Bor el is true. It is of interest to observe that while the Hedrick property, || the derived set of every set is closed, may not be present in a topological space, it is present in associated spaces discussed in § §8 and 12. This fact permits the generalization to general topological spaces of many classical theorems. The second part of the paper is devoted to a study of covering theorems in general topological space.If Of particular interest is the formulation in §20 of two sets of necessary and sufficient conditions that a set possess the "any to finite" form of the property of Borel.