A Generalized Borsuk-Ulam Theorem which is Equivalent to Some Propositions

Abstract:

      In this paper, we prove a generalized form of Borsuk-Ulam theorem: Generalized Borsuk-Ulam Theorem Let (X, T) be T-space, where X is a n-dimensional homology sphere over the group T₂ of integers mod 2, T is a topological transformation on X weth period 2 and hasn't any fixed points; f: X (?) Rⁿ is any con-tinuous map from X into n-euclidean space Rn then, there existe at least a pair (X, T(X)) of involution points of some x ∈ X mapping into one point, i.e. f(X) = f(T(X)). In addition, we show that it is equivalent to some propositions too.