On the Euler characteristic of spherical polyhedra and the Euler relation

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Let Eⁿ⁺¹, for some integer n ≥ 0, be the (n + 1)-dimensional Euclidean space, and denote by Sⁿ the standard n–sphere in Eⁿ⁺¹, . It is convenient to introduce the (–1)-dimensional sphere , where denotes the empty set. By an i-dimensional subsphere T of Sⁿ, i = 0 n, we understand the intersection of Sⁿ with some (i+1)-dimensional subspace of Eⁿ⁺¹. The affine hull of T always contains, with this definition, the origin of Eⁿ⁺¹. is the unique (–1)-dimensional subsphere of Sⁿ. By the spherical hull, sph X, of a set , we understand the intersection of all subspheres of Sⁿ containing X. Further we set dim X: = dim sph X. The interior, the boundary and the complement of an arbitrary set , with respect to Sⁿ, shall be denoted by int X, bd X and cpl X. Finally we define the relative interior rel int X to be the interior of with respect to the usual topology sphZ . For each (^n–1)-dimensional subsphere of Sⁿ defines two closed hemispheres of Sⁿ, whose common boundary it is. The two hemispheres of the sphere Sº are denned to be the two one-pointed subsets of Sº. A subset is called a closed (spherical) polytope, if it is the intersection of finitely many closed hemispheres, and, if, in addition, it does not contain a subsphere of Sⁿ. is called an i-dimensional, relatively open polytope, , or shortly an i-open polytope, if there exists a closed polytope such that dim P = i and Q = rel int P. is called a closed polyhedron, if it is a finite union of closed polytopes P¹ …, P^r. The empty set is the only (–1)-dimensional closed polyhedron of Sⁿ. We denote by the set of all closed polyhedra of Sⁿ. is called an i-open polyhedron, for some , if there are finitely many i-open polytopes Q¹ …, Q^r in Sⁿ such that , and dim . By we denote the set of all i-open polyhedra. Clearly for all , and each i-dimensional subsphere of Sⁿ, , belongs to and to , For each i-dimensional subsphere T of Sⁿ, set . A map is defined by , for all , and, for all .