Abstract
This paper presents an exposition of the topological foundations of the theory of n-valued maps. By means of proofs that exploit particular features of n-valued functions, as distinct from more general classes of multivalued functions, we establish, among other properties, the equivalence of several definitions of continuity. The exposition includes an exploration of the role of configuration spaces in the study of n-valued maps. As a consequence of this point of view, we extend the classical Splitting Lemma, that is central to the fixed point theory of n-valued maps, to a characterization theorem that leads to a new type of construction of non-split n-valued maps.
Highlights
An n-valued map φ : X Y is a continuous multivalued function that associates to each x ∈ X an unordered subset of exactly n points of Y
The fixed point theory of n-valued maps φ : X X has been a topic of considerable interest in recent years and there is much research activity at the present time [3] - [12], [15], [16]
The purpose of this paper is to present the topological foundations of this subject
Summary
An n-valued map φ : X Y is a continuous multivalued function that associates to each x ∈ X an unordered subset of exactly n points of Y. The Nielsen fixed point theory of an n-valued map φ : X X presented by Schirmer in [23] - [25] depends on a version of the Splitting Lemma which she obtained from a more general result concerning multivalued functions on connected compact metric spaces due to O’Neill [21]. Another consequence of the Splitting Lemma, in [6], is that an n-valued map φ : X Y of finite simplicial complexes induces a chain map of their simplicial chain complexes. The authors thank Ofelia Alas for clarifying for them the relationship of the continuity of an n-valued function and the continuity of a mapping to the corresponding configuration space
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
