In the beginning of the 20th century, A. N. White-
head [39, 40] and T. de Laguna [9] proposed a new theory of space,
known as region-based theory of space. They did not present their ideas
in a detailed mathematical form. In 1997, P. Roeper [33] has shown
that the locally compact Hausdorff spaces correspond bijectively (up to
homeomorphism and isomorphism) to some algebraical objects which
represent correctly Whitehead’s ideas of region and contact relation,
generalizing in this way a previous analogous result of de Vries [10] con-
cerning compact Hausdorff spaces (note that even a duality for the cate-
gory of compact Hausdorff spaces and continuous maps was constructed
by de Vries [10]). Recently, a duality for the category of locally compact
Hausdorff spaces and continuous maps, based on Roeper’s results, was
obtained in [11] (it extends de Vries’ duality mentioned above). In this
paper, using the dualities obtained in [10, 11], we construct directly (i.e.
without the help of the corresponding topological spaces) the dual objects
of Euclidean spaces, spheres, tori and Tychonoff cubes; these algebraical
objects completely characterize the mentioned topological spaces. Thus,
a mathematical realization of the original philosophical ideas of White-
head [39, 40] and de Laguna [9] about Euclidean spaces is obtained.
