the importance of the following Problem. Let A : X—*Y be a mapping (not necessarily linear) of a topological space X into a topological space Y. Under what conditions is A (X) open in F? The aim of this paper is to give a particular solution of this problem in the case of mappings A : X—>X of a Banach space X into itself. It will be shown that the Fixed Point Theorems of Schauder and Brouwer may be applied to find conditions under which the image A(X) of X is open in X. The idea of the following proofs is: Suppose that A: X-+X is a mapping of X into itself and let yoŒA(X). To prove that A (X) contains a spherical region S(ya, r
By a result of Mazurkiewicz and Sierpinski, there exist N, topological types of compact and countable sets.' Since a countable set is 0-dimensional, there arises a natural question: what is the power of topological types of other classes of 0-dimensional sets? In this paper we consider separable metric spaces only. Every 0-dimensional space being topologically contained in the Cantor set2 C, we confine ourselves to subsets of this set. We prove the following three theorems:
