DECOMPOSITIONS OF A Ti SPACE

Introduction. Several authors have proved theorems of the type: The "structure" of a certain class of transformations defined on a "suitable" space A to a fixed "suitable" space B "determines" the space A. As examples, we have: Banach [3, p. 170, see also 6, 7, 13] i The Banach space of all real, continuous functions defined on a compact metric space A "determines" A. Eidelheit [5, see also 2, 10, l l ] : The ring of all bounded operators on a real Banach space A "determines" A. In the present paper, we prove an analogous theorem (Theorem 2.5). Intuitively, it says that a T\ space A is "determined" by a rather weak ordered system structure of the collection of all continuous mappings of A onto an arbitrary (variable) T\ space B. More exactly, it states: If two 7\ spaces A, B are such that the ordered system of upper semi-continuous decompositions of A is isomorphic to that of By then A and B are homeomorphic. In §1 we give a discussion of ordered systems which is sufficient for our purposes. In §2 we prove the theorem mentioned above. In §3 we characterize separation and connectedness properties of a Ti space in terms of order properties of its upper semi-continuous decompositions. In §4 we discuss compactness properties of the space and their relations to order properties of the decompositions, and in §5 we give some examples and counter examples.