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2000/2001 Setwise Quasicontinuity and Π-Related Topologies
Annalisa Crannell, Ralph Kopperman
Real Anal. Exchange 26(2): 609-622 (2000/2001).


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


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Annalisa Crannell. Ralph Kopperman. "Setwise Quasicontinuity and Π-Related Topologies." Real Anal. Exchange 26 (2) 609 - 622, 2000/2001.


Published: 2000/2001
First available in Project Euclid: 27 June 2008

zbMATH: 1011.54033
MathSciNet: MR1844140

Primary: 37B99 , 54H20‎

Keywords: $\Pi$-related topologies , $\pi\sp o$-base , semi-open set , separate quasicontinuity , setwise , topological transitivity

Rights: Copyright © 2000 Michigan State University Press

Vol.26 • No. 2 • 2000/2001
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