Some Theorems on Colocally Connected Continua
Keywords:
colocally connected, proximately refinable map, refinable map, Whitney property, Whitney reversible propertyAbstract
We show that each refinable map preserves colocal
connectedness of the domain while a proximately refinable map does not necessarily. Also, we prove that colocal connectedness is a Whitney property and is not a Whitney reversible property.
References
Idalia-Guadalupe Bautista-Callejas, Mauricio Chacón-Tirado, and Raúl Escobedo, Non-weak cut, shore and non-cut points in Whitney levels, Topology Appl. 283 (2020), 107335, 6 pp.
Jozef Bobok, Pavel Pyrih, and Benjamin Vejnar, Non-cut, shore and non-block points in continua, Glas. Mat. Ser. III 51(71) (2016), no. 1, 237-253.
Benjamin Espinoza and Eiichi Matsuhashi, $D$-continua, $D^*$-continua, and Wilder continua, Topology Appl. 285 (2020), 107393, 25 pp.
Jo Ford and J. W. Rogers, Jr., Refinable maps, Colloq. Math. 39 (1978), no. 2, $263-269$.
Jack T. Goodykoontz, Jr. and Sam B. Nadler, Jr., Whitney levels in hyperspaces of certain Peano continua, Trans. Amer. Math. Soc. 274 (1982), no. 2, 671-694.
E. E. Grace, Generalized refinable maps, Proc. Amer. Math. Soc. 98 (1986), no. 2, 329-335.
E. E. Grace and E. J. Vought, Proximately refinable maps and $\theta_n$-continua, Topology Proc. 15 (1990), 39-51.
Hiroshi Hosokawa, Aposyndesis and coherence of continua under refinable maps, Tsukuba J. Math. 7 (1983), no. 2, 367-372.
Alejandro Illanes, and Sam B. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances. Monographs and Textbooks in Pure and Applied Mathematics, 216. New York: Marcel Dekker, Inc., 1999.
F. Burton Jones, Aposyndetic continua and certain boundary problems, Amer. J. Math. 63 (1941), 545-553.
Ivan Lončar, Weight and metrizability of inverses under hereditarily irreducible mappings, An. Ştiinţ. Univ. "Ovidius" Constanţa Ser. Mat. 16 (2008), no. 2, 6781.
Sergio Macías, Topics on Continua. Boca Raton, FL: Chapman \& Hall/CRC, 2005.
Sam B. Nadler, Jr. Continuum Theory. An Introduction. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 158. New York: Marcel Dekker, Inc., 1992.
Ann Petrus, Whitney maps and Whitney properties of $C(X)$, Topology Proc. 1 (1977), 147-172.
