Mathematics – General Topology
Scientific paper
2012-03-19
Mathematics
General Topology
29 pages
Scientific paper
In the beginning of the 20th century, A. N. Whitehead and T. de Laguna proposed a new theory of space, known as {\em region-based theory of space}. They did not present their ideas in a detailed mathematical form. In 1997, P. Roeper 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 {\em region} and {\em contact relation}, generalizing in this way a previous analogous result of de Vries concerning compact Hausdorff spaces (note that even a duality for the category of compact Hausdorff spaces and continuous maps was constructed by de Vries). Recently, a duality for the category of locally compact Hausdorff spaces and continuous maps, based on Roeper's results, was obtained by G. Dimov (it extends de Vries' duality mentioned above). In this paper, using the dualities obtained by de Vries and Dimov, 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 Whitehead and de Laguna about Euclidean spaces is obtained.
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