Continuous Images of a Line Segment

Abstract

With squares and triangles and all their continuous images revealed as continuous images of the interval I, the question arose as to the general characterization of such sets. In 1908, A. Schoenflies found such a characterization (see Schoenflies [1], p. 237), which, by its very nature, only applies to sets in the two-dimensional plane. In 1913, Hans Hahn and Stefan Mazurkiewicz independently arrived at a complete characterization of such sets in Eⁿ (Hahn [2], [3], Mazurkiewicz [1], [2], [3]). Their results can be extended to apply to even more general spaces, as we will point out at the end of Section 6.8. We will follow Hahn’s approach, commenting on Mazurkiewicz’ work briefly in Section 6.7. The development of this chapter may be viewed as a generalization of Lebesgue’s construction of a space-filling curve. Two elements made this construction possible: First, the square Q emerged as a continuous image of the Cantor set and, secondly, it was possible to extend the definition of the mapping continuously from Γ into I by linear interpolation, i.e., by joining the image of the left endpoint of an interval that has been removed in the construction of the Cantor set to the image of the right endpoint by a straight line (which lies in the square Q). Hausdorff has shown that every compact set is a continuous image of the Cantor set. But compactness is not enough to ensure that the above mentioned images can be joined by a continuous arc that remains in the set in such a manner that the extended map is continuous.