Omega-Limit Sets and Non-Continuous Functions

Mimna, Roy A.

doi:10.2307/44152815

Cited by 1 publication

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“…Mimna [14] recently investigated omega-limit sets and the dense mapping property (DMP) for quasicontinuous functions f : R → R. We claim that the extension from continuous systems to quasicontinuous systems is a natural one; moreover it can easily build on the topological and analytical groundwork which has already been laid.…”

Section: Introductionmentioning

confidence: 99%

Setwise Quasicontinuity and Π-Related Topologies

Crannell¹,

Kopperman²

2000

Real Analysis Exchange

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A function is quasicontinuous if inverse images of open sets are semiopen. 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 Π-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 Π-related to it, and with respect to which each of the functions is continuous to the range space.Further, the set of iterates {1X , f, f •f, . . . } of a self-map f : X → X, is setwise quasicontinuous if and only if the topology can be extended to a Π-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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Section: Introductionmentioning

confidence: 99%