A Whiteheadian-type description of Euclidean spaces, spheres, tori and Tychonoff cubes

Abstract

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.