## Real Analysis Exchange

- Real Anal. Exchange
- Volume 26, Number 2 (2000), 609-622.

### Setwise Quasicontinuity and Π-Related Topologies

Annalisa Crannell and Ralph Kopperman

#### 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

#### Article information

**Source**

Real Anal. Exchange, Volume 26, Number 2 (2000), 609-622.

**Dates**

First available in Project Euclid: 27 June 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1214571354

**Mathematical Reviews number (MathSciNet)**

MR1844140

**Zentralblatt MATH identifier**

1011.54033

**Subjects**

Primary: 37B99: None of the above, but in this section 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

**Keywords**

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

#### Citation

Crannell, Annalisa; Kopperman, Ralph. Setwise Quasicontinuity and Π-Related Topologies. Real Anal. Exchange 26 (2000), no. 2, 609--622. https://projecteuclid.org/euclid.rae/1214571354