Atomic mappings and extremal continua
- Autores: Janusz J. Charatonik
- Localización: Extracta mathematicae, ISSN-e 0213-8743, Vol. 7, Nº 2-3, 1992, págs. 131-134
- Idioma: inglés
- Enlaces
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Resumen
The notion of atomic mappings was introduced by R. D. Anderson in [1] to describe special decompositions of continua. Soon, atomic mappings turned out to be important tools in continuum theory. In particular, it can be seen in [2] and [5] that these maps are very helpful to construct some special, singular continua. Thus, the mappings have proved to be interesting by themselves, and several of their properties have been discovered, e.g. in [6], [7] and [9]. The reader is referred to Table II of [9], p. 28, where relations between atomic and other classes of mappings are exhibited. The paper is devoted to further study of various properties of atomic mappings. As the first main result of the paper it is shown that if a mapping between metric continua is atomic, then the inverse image of an irreducible subcontinuum of the range is an irreducible subcontinuum of the domain (Theorem 2). This result is then applied to investigate behaviour of extremal continua under atomic mappings. The concept of extremal continuum has been introduced by M. A. Owens in [10] as a special kind of terminal continuum. Mapping invariance of these continua was studied in the author's paper [3] (see also [4] for a related topic). The second main result of the paper supplies that study. Namely, it is shown that if a surjective mapping between continua is atomic, then the image of an extremal subcontinuum in the domain is an extremal subcontinuum in the range (Corollary 5).
