Open and closed maps

About: Open and closed maps is a research topic. Over the lifetime, 658 publications have been published within this topic receiving 7384 citations.

Contra-continuous functions and strongly S-closed spaces

Abstract: In 1989 Ganster and Reilly [6] introduced and studied the notion of LC-continuous functions via the concept of locally closed sets. In this paper we consider a stronger form of LC-continuity called contra-continuity. We call a function f:(X,τ)→(Y,σ) contra-continuous if the preimage of every open set is closed. A space (X,τ) is called strongly S-closed if it has a finite dense subset or equivalently if every cover of (X,τ) by closed sets has a finite subcover. We prove that contra-continuous images of strongly S-closed spaces are compact as well as that contra-continuous, β-continuous images of S-closed spaces are also compact. We show that every strongly S-closed space satisfies FCC and hence is nearly compact.

On semiconjugation of entire functions

Abstract: Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g∘f=h∘g. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g−1(J(h))=J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that g∘f=h∘g.

A note on closed maps and compact sets

Abstract: Some results relating closed maps to compact sets, which are already known for metrizable spaces, are here proved in a more general setting. Examples are given to indicate barriers to further improvements.

Locally closed sets and LC-continuous functions

Abstract: In this paper we introduce and study three different notions of generalized continuity, namely LC-irresoluteness, LC-continuity and sub-LC-continuity. All three notions are defined by using the concept of a locally closed set. A subset S of a topological space X is locally closed if it is the intersection of an open and a closed set. We discuss some properties of these functions and show that a function between topological spaces is continuous if and only if it is sub-LC-continuous and nearly continuous in the sense of Ptak. Several examples are provided to illustrate the behavior of these new classes of functions.