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ISSN : 1226-0657

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Vol.20 No.1

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ON ARCWISE CONNECTEDNESS IM KLEINEN IN HYPERSPACES

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081
2013, v.20 no.1, pp.71-78
https://doi.org/10.7468/jksmeb.2013.20.1.71
Baik, Bong Shin
Rhee, Choon Jai
Baik,, B. S. , & Rhee,, C. J. (2013). ON ARCWISE CONNECTEDNESS IM KLEINEN IN HYPERSPACES. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 20(1), 71-78, https://doi.org/10.7468/jksmeb.2013.20.1.71

Abstract

Let X be a space and <TEX>$2^X$</TEX>(C(X);K(X);<TEX>$C_K$</TEX>(X)) denote the hyperspace of nonempty closed subsets(connected closed subsets, compact subsets, subcontinua) of X with the Vietoris topology. We investigate the relationships between the space X and its hyperspaces concerning the properties of connectedness im kleinen. We obtained the following : Let X be a locally compact Hausdorff space. Let <TEX>$x{\in}X$</TEX>. Then the following statements are equivalent: (1) X is connected im kleinen at <TEX>$x$</TEX>. (2) <TEX>$2^X$</TEX> is arcwise connected im kleinen at {<TEX>$x$</TEX>}. (3) K(X) is arcwise connected im kleinen at {<TEX>$x$</TEX>}. (4) <TEX>$C_K$</TEX>(X) is arcwise connected im kleinen at {<TEX>$x$</TEX>}. (5) C(X) is arcwise connected im kleinen at {<TEX>$x$</TEX>}.

keywords
hyperspace, connected im kleinen, arcwise connected im kleinen

Reference

1.

(1979). Local connectedness, connectedness im Kleinen and other properties of hyperspaces of <TEX>$R_0$</TEX> Spaces. Math. Vesnik, 3, 113-123.

2.

(1974). Connectedness im Kleinen and local connectedness in <TEX>$2^X$</TEX> and C(X). Pacific J. of Math., 53, 387-397. 10.2140/pjm.1974.53.387.

3.

(1998). Local properties of hyperspaces (183-200). Topology Proceedings.

4.

(1968). Arcs, Semigroups, and Hyperspaces. Canadian J. Math., 20, 1207-1210. 10.4153/CJM-1968-115-3.

5.

(1951). Topology on spaces of subsets. Trans. Amer. Math. Soc., 71, 152-182. 10.1090/S0002-9947-1951-0042109-4.

6.

(1990). C-supersets, piecewise order-arcs and local arcwise connectedness in hyperspaces. Q & A in General Topology, 8, 467-485.

7.

(1922). Bereiche Zweiter Ordung. Monatshefte fur Mathematik und Physik, 32, 253-280.

8.

(1923). Kontinua Zeiter Ordung. Monatshefte fur Mathematik und Physik, 33, 49-62. 10.1007/BF01705590.

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

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