The Stein Characterization Of Open Sets Which Satisfy The Oka Principle In A Infinite Didensional Space

Similar to the definition of holomorphic functions through Cauchy integral formulain the theory of functions of one complex variable, Using the Cauchy integral formula,we can also give the definition of holomorphic functions of several variables. In thetheory of functions of several variables, zero set of a holomorphic function is notisolated, then we have the concept of holomorphic domain, that is the largest domain ofa holomorphic functions.Cartan first construct the holomorphic hull for the characterization of holomorphicdomain, but more importantly is how to characterize the geometric features of aholomorphic domain, through the introduction of the concept of pseudoconvex domain,we got that holomorphic domain is equal to pseudoconvex domain.Because the nature of holomorphic functions are largely determined by thetopological and geometric properties of the defined area, therefore, the focus of thestudy experienced a shift from the local nature to the whole nature. To achieve this aimwe mainly use the sheaf and the cohomology theory as a tool, with the development andsolution of the Cousin problem, the sheaf theory and the cohomology theory of thesheaf started as a tool to character the domain of holomorphy, and got a wealth ofdevelopment.The aim of this paper is to expanded M.Abe’s conclusion about thecharacterization of open set which satisfy the Oka-Grauert principle in a Stein space,combining the thought of Xiaodong Li[15] in characterizing the open subset of a infiniteStein manifold to the infinite dimensional space with finite open topology.And similarly, to get the generalization of the open set which satisfy the Oka-Grauert principle in a Stein space with some specific properties to the infinitedimensional space with finite open topology.