Cyclic Invariance under Multi-valued Maps

When ƒ is single-valued we know that continuity is equivalent to the assertion that A, B closed imply f (A), f~(B) closed. When ƒ is multi-valued we take this as a definition of continuity. I t does not follow, as in the single-valued case, that f~~(B) is open if B is open. These definitions include both a single-valued map ( = continuous function) and its inverse. In this note we show that certain theorems of analytic topology carry over to multi-valued maps ( = continuous multi-valued functions as defined above). Some of our results are new even for singlevalued maps. Except for fixed-point theorems there seem to be no results in the literature for multi-valued maps. We say that ƒ is anarthric if it is continuous and if for y Ci Y no xÇzX—f~{y) separates f~"{y) in X. If ƒ is single-valued and nonalternating, then ƒ is anarthric. See Wallace [2], [3], and [4] and Whyburn [5] and [ó]. I t is clear that if ƒ is the inverse of a singlevalued map, then ƒ is anarthric. For simplicity we write P \ Q to mean that the sets P and Q are mutually separated. Also if py qG.X, then p~q means that no point separates p and q in X.