Setwise Quasicontinuity and Π-Related Topologies

Abstract

A function is quasicontinuous if inverse images of open sets are semi-open. We generalize this definition: a collection of functions is setwise quasicontinuous if finite intersections of inverse images of open sets by functions in the collection are semi-open (so a function is quasicontinuous if and only if its singleton is a setwise quasicontinuous set). Two topologies on the same space are $\Pi$-related if each nonempty open set (in each) has non-empty interior with respect to the other. This paper demonstrates that a dynamical system is setwise quasicontinuous if and only if the original topology can be strengthened to one which is $\Pi$-related to it, and with respect to which each of the functions is continuous to the range space. Further, the set of iterates $\{1_X,f,f\circ f,\dots\}$ of a self-map $f:X\to X$, is setwise quasicontinuous if and only if the topology can be extended to a $\Pi$-related one, so that each iterate is continuous from the new space to the new space. We present a quasicontinuous function on the unit interval which is discontinuous on a dense subset of the interval; and show that conjugacies of dynamical systems via quasicontinuous bijections preserve much of the desired structure of the systems