Some properties of limited continua, irreducible between two points

Abstract

1. In reading papers on the theory of point aggregates one frequently meets with the continuum C whose points are defined as follows: when -1?! x < O or O < x < 1, y = sin (l/x); when x=O, -1 < y < 1. This is perhaps the most commonly cited example of a limited continuum irreducible between two points and having a continuum of condensation. If we regard the concept of connectedness im kleinen as the analogue for continua of continuity in functions, we notice that the properties of this continuum resemble those of pointwise discontinuous functions. The points where the continuum is connected im kleinen form a set of the secondary category with respect to C, while those of the second genre t form a set of the first category. The question at once arises as to whether this similarity is of a general character or is merely due to the nature of the example cited, and suggests that a study of limited continua irreducible between two points with special reference to the oscillation at the various points would be of interest. This problem has been discussed in a paper by H. Hahn,t who has shown that such a continuum is the sum of a set of sub-continua known as parts, no two of which have common points. However, the fact that in many cases a prime part itself can be subdivided indicates that the subject has not been exhausted and it is the purpose of this paper to present some further results along this line. The first half of the paper (??4-15) is devoted to the general properties of the oscillation of a limited irreducible continuum. In the second half the properties such continua have when the points of the first genre are everywhere dense are treated, and in particular it is shown that in this case there is a correspondence between the points of such a continuum and those of a linear segment analogous to that between the variables y and x when y =f(x) is a pointwise discontinuous function of a certain type.