9f4849df-ae01-4f58-ba32-8c87af3e075320240227080634037wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40511220241220242310.37394/23206.2024.23https://wseas.com/journals/mathematics/2024.phpMaysoonQousiniFaculty of Science and Information Technology, Al-Zaytoonah University of Jordan, 11183, Amman, JORDANHasanHdiebFaculty of Science, The University of Jordan, 11942, Amman, JORDANEmanAlmuhurDepartment of Basic Science and Humanities, Applied Science Private University, 11931, Amman, JORDANThe study of applications of locally compact spaces in polyhedra in relation to their dimensions as well as homotopy and extension problems developed in the late 1940s and early 1950s under the leadership of mathematician. Many mathematicians studied application locally compact in polyhedra. A polyhedron can be obtained by subdivision, as a simplicial metric complex; thus, re-gluings of polyhedra can also be seen as simple complexes. Thus, the topology of a simplicial metric complex X is the topology quotient of the reattachment. The objective of this work is to shed light on the applications in polyhedra of locally compact spaces and to highlight the limits of these applications. A continuous application f of X in P defines a finite open overlay of X, and a partition of the unit subordinate to this overlay, f is homotopic to an application f ', obtained by composing the restriction to A, of an application of X in the KR polyhedron, and a simplistic application of a sub-polyhedron KR' in P. The problem of extension deserves to be elucidated to understand how it is possible to get around certain conceptual difficulties.2272024227202411812414https://wseas.com/journals/mathematics/2024/a285106-1859.pdf10.37394/23206.2024.23.14https://wseas.com/journals/mathematics/2024/a285106-1859.pdf10.2307/1969550Deane Montgoemery, “Theorems on the Topological Structure of Locally Compact Groups,” in Annals of Mathematics, vol. 50, no. 3, pp. 570-580, 1949. 10.5486/pmd.1998.1780B. Cervar, and N. Uglesi, “Paracompact locally compact spaces as inverse limits of polyhedral,” Publ. Math. Debrecen, vol. 52, pp.69-78, 1998 10.1090/s0002-9947-1938-1501971-xDunford, N, “Uniformity in linear spaces,” in Transactions of the American Mathematical Society, vol. 44, no. 3, pp. 305-356, 1938, DOI 10.1090/S0002-9947-1938-1501971-X. Banach, S, “banach spaces, compact operators, linear functionals,” in Theory of linear operations, 1st ed, Varsovie, 1932, 49-99. 10.2307/1989960Dunford, N, Pettis, J, “Linear transformations on summable fonctions,” in Transactions of the American Mathematical Society, vol. 47, no. 3, pp. 323-392, 1940. Philipps, R, “On linear transformations,” in Transactions of the American Mathematical Society, vol. 48, no. 3, pp. 516-541, 1940. Stone, M, “Applications of the theory of Booleans rings to general topology,” in Transactions of the American Mathematical Society, vol. 41, no. 3, pp. 375-481, 1973.