Baik, Bong Shin;Rhee, Choon Jai

  • Received : 2014.11.10
  • Accepted : 2014.12.01
  • Published : 2014.12.25


We investigate the relationships between the space X and the hyperspaces concerning admissibility and connectedness im kleinen. The following results are obtained: Let X be a Hausdorff continuum, and let A, $B{\in}C(X)$ with $A{\subset}B$. (1) If X is c.i.k. at A, then X is c.i.k. at B if and only if B is admissible. (2) If A is admissible and C(X) is c.i.k. at A, then for each open set U containing A there is a continuum K and a neighborhood V of A such that $V{\subset}IntK{\subset}K{\subset}U$. (3) If for each open subset U of X containing A, there is a continuum B in C(X) such that $A{\subset}B{\subset}U$ and X is c.i.k. at B, then X is c.i.k. at A. (4) If X is not c.i.k. at a point x of X, then there is an open set U containing x and there is a sequence $\{S_i\}^{\infty}_{i=1}$ of components of $\bar{U}$ such that $S_i{\longrightarrow}S$ where S is a nondegenerate continuum containing the point x and $S_i{\cap}S={\emptyset}$ for each i = 1, 2, ${\cdots}$.


hyperspace;connected im kleinen;admissibility


  1. J. T. Goodykoontz, Jr., More on connectedness im kleinen and Local Connectedness in C(X), Proc. Amer. Math. Soc. 65 (1977), 357-364.
  2. J. T. Goodykoontz, Jr., Local arcwise $2^X$ and C(X), Houston J. Math. 4 (1978), 41-47.
  3. J. T. Goodykoontz, Jr. and C. J. Rhee, Local properties of hyperspaces, Topology Proceedings 23 (1998), 183-200.
  4. K. Kuratowski, Topology, Vol.II, Warszawa, 1968.
  5. E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.
  6. W. Makuchowski, On Local Connectedness in Hyperspaces, Bull. Polish. Acad. Sci. Math. 47(2) (1999), 119-126.
  7. W. Makuchowski, On Local Connectedness at a Subcontinuum and Smoothness of Continua, Houston J. Math. 4(3) (2003), 711-716.
  8. D. E. Bennett and J. B. Fugate, Continua and their non-separating subcontinua, Dissertationes Math. (Rozprawy Mat.) 149 (1977), 1-46.
  9. J. T. Goodykoontz, Jr., Connectedness im kleinen and local connectedness in $2^X$ and C(X), Pacific J. Math. 53 (1974), 387-397.
  10. C. J. Rhee, Obstucting sets for hyperspace, Topology Proceedings 15 (1985), 159-173.
  11. G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publication, 28 (1942).
  12. M. Wojdyslawski, Retract absolus et hyperspaces des continus, Fund. Math. 32 (1939), 184-192.
  13. B. S. Baik, Adimssible fibers and Ri-sets, Korean Annals of Math. 16 (1999), 177-182.
  14. B. S. Baik, A Note on Connectedness Im Kleinen in C(X), Submitted.