Linear Invariance of Lindeloef Numbers

摘要

Let X and Y be Tychonov spaces and suppose there exists a continuous linearbijection from C(sub P)(X) to C(sub p)(Y). In this paper the authors develop a method that enables them to compare the Lindelof number of Y with the Lindelof number of some dense subset Z of X. As a corollary the authors get that if for perfect spaces X and Y, C(sub p)(X) and C(sub p)(Y) are linearly homeomorphic, then the Lindelof numbers of X and Y are equal. Another result in the paper is the following. Let X and Y be any two zero dimensional metric spaces or any two linearly ordered perfect Tychonov spaces such that C(sub p)(X) and C(sub p)(Y) are linearly homeomorphic. Let P be a topological property that is closed hereditary, closed under taking countable unions and closed under taking continuous images. Then X has property P is and only if Y has. As examples of such properties the authors consider certain cardinal functions.