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|Title:||A Whiteheadian-type description of Euclidean spaces, spheres, tori and Tychonoff cubes||Authors:||Dimov, Georgi D.||Keywords:||Euclidean spaces; Tychonoff cubes; spheres; tori; (locally) compact Hausdorff spaces; duality; regular closed sets; sums of local contact algebras; sums of normal contact algebras||Issue Date:||2012||Publisher:||EUT Edizioni Università di Trieste||Source:||Georgi D. Dimov, "A Whiteheadian-type description of Euclidean spaces, spheres, tori and Tychonoff cubes", in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 44 (2012), pp. 45–74.||Series/Report no.:||Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
|Abstract:||In the beginning of the 20th century, A. N. White- head [39, 40] and T. de Laguna  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  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  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 ). Recently, a duality for the category of locally compact Hausdorff spaces and continuous maps, based on Roeper’s results, was obtained in  (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  about Euclidean spaces is obtained.||URI:||http://hdl.handle.net/10077/8273||ISSN:||0049-4704|
|Appears in Collections:||Rendiconti dell'Istituto di matematica dell'Università di Trieste: an International Journal of Mathematics vol.44 (2012)|
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