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Keywords:
continuum; symmetric product; quotient space; hyperspace; induced mapping
continuum; symmetric product; quotient space; hyperspace; induced mapping
Summary:
Given a metric continuum $X$ and a positive integer $n$, $F_{n}(X)$ denotes the hyperspace of all nonempty subsets of $X$ with at most $n$ points endowed with the Hausdorff metric. For $K\in F_{n}(X)$, $F_{n}(K,X)$ denotes the set of elements of $F_{n}(X)$ containing $K$ and $F_{n}^K(X)$ denotes the quotient space obtained from $F_{n}(X)$ by shrinking $F_{n}(K,X)$ to one point set. Given a map $f\colon X\to Y$ between continua, $f_{n}\colon F_{n}(X)\to F_{n}(Y)$ denotes the induced map defined by $f_{n}(A)=\nobreak f(A)$. Let $K\in F_{n}(X)$, we shall consider the induced map in the natural way $f_{n,K}\colon F_{n}^K(X)\to F_{n}^{f(K)}(Y)$. In this paper we consider the maps $f$, $f_{n}$, $f_{n,K}$ for some $K\in F_n(X)$ and $f_{n,K}$ for each $K\in F_n(X)$; and we study relationship between them for the following classes of maps: homeomorphisms, monotone, confluent, light and open maps.
References:
[1] Barragán F.: Induced maps on $n$-fold symmetric product suspensions. Topology Appl. 158 (2011), no. 10, 1192–1205. DOI 10.1016/j.topol.2011.04.006 | MR 2796121
[2] Castañeda-Alvarado E., Mondragón R. C., Ordoñez N., Orozco-Zitli F.: The hyperspace $F_n^K(X)$. Bull. Iranian Math. Soc. 47 (2021), no. 3, 659–678. MR 4249170
[3] Dugundji J.: Topology. Allyn and Bacon, Boston, 1966. MR 0193606 | Zbl 0397.54003
[4] Higuera G., Illanes A.: Induced mappings on symmetric products. Topology Proc. 37 (2011), 367–401. MR 2740654
[5] Hosokawa H.: Induced mappings between hyperspaces II. Bull. Tokyo Gakugei Univ. (4) 44 (1992), 1–7. MR 1193338
[6] Kuratowski K.: Topology. Academic Press, New York, London, Państwowe Wydawnictwo Naukowe, Warsaw, 1968. Zbl 0849.01044
[7] Macías S.: Aposyndetic properties of symmetric products of continua. Topology Proc. 22 (1997), 281–296. MR 1657883
[8] Macías S.: Topics on Continua. Pure Appl. Math. Ser., 275, Chapman and Hall/CRC, Taylor and Francis Group, Boca Raton, 2005. MR 2147759
[9] Maćkowiak T.: Continuous Mappings on Continua. Dissertationes Math., Rozprawy Mat., 158, 1979. MR 0522934
[10] Nadler S. B., Jr.: Hyperspaces of Sets. A Text with Research Questions, Monographs and Texbooks in Pure and Applied Mathematics, 49, Marcel Dekker, New York-Basel, 1978. MR 0500811 | Zbl 1125.54001
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