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

The abstract of the talk (see below) may be downloaded in one of the following formats:

We consider the problem of partitioning the real plane --- or suitable subsets
of it --- into (non-degenerated) closed segments. In particular, we show that:
{\it
\begin{enumerate}
\item
It is possible to find a partition of $\R^2$ into closed segments, all having
the same lenght.
\item
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.
\end{enumerate}
}
Let also call a subset of the real plane {\it segment-free}, if it contains no
non-degenerated 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.

\begin{thebibliography}{9}

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K.~Ciesielski,
{\em Set theory for the working mathematician\/} (\S\,6.1),
London Mathematical Society Student Texts, 39. Cambridge University Press,
Cambridge, 1997.

\bibitem{Erdos69}
P.~\H Erdos,
{\em Problems and results in chromatic graph theory},
Proof techniques in graph theory (F. Harary, ed.), Academic Press, New York,
1969, 27--35.

\bibitem{Sie19}
W.~Sierpinski,
{\em Sur un th\'eor\eme \'equivalent \a l'hypoth\ese du continu},
Bull. Int. Acad. sci. Cracovie {\bf A} (1919), 1--3.
\end{thebibliography}

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