ITES2001 Fourth Italian-Spanish conference on General Topology and its applications Bressanone, 27-30 June 2001

Pairwise disjoint segments in the real plane

Camillo Costantini Dipartimento di Matematica, Università di Torino, Italy

We consider the problem of partitioning the real plane — or suitable subsets of it — into (non-degenerated) closed segments. In particular, we show that:

(1) It is possible to find a partition of ℝ^{2} into closed segments, all having the same lenght.

(2) It is possible to find a partition of ℝ^{2} into closed segments, each of which is parallel to either the first or the second co-ordinate axis. Moreover, we may require each element of such a partition to have a lenght chosen among two pre-assigned ones.

Let also call a subset of the real plane segment-free, if it contains no nondegenerated closed segment. We prove that there is a subset of the real plane, with segment-free complement, which may be partitioned into countably many closed segments. The construction is based on the idea of creating inductively an infinite grill, by adding at each stage segments of decreasing lenght, which are alternately parallel to the first or the second co-ordinate axis.

References

[1] K. Ciesielski, Set theory for the working mathematician (§ 6.1), London Mathematical Society Student Texts, 39. Cambridge University Press, Cambridge, 1997.

[2] P. E̋rdos, Problems and results in chromatic graph theory, Proof techniques in graph theory (F. Harary, ed.), Academic Press, New York, 1969, 27–35.

[3] W. Sierpinski, Sur un théorème équivalent à l’hypothèse du continu, Bull. Int. Acad. sci. Cracovie A (1919), 1–3.