ON SUBMAXIMAL AND QUASI-SUBMAXIMAL SPACES

• Journal title : Honam Mathematical Journal
• Volume 32, Issue 4,  2010, pp.643-649
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2010.32.4.643
Title & Authors
ON SUBMAXIMAL AND QUASI-SUBMAXIMAL SPACES
Lee, Seung-Woo; Moon, Mi-Ae; Cho, Myung-Hyun;

Abstract
The purpose of this paper is to study some properties of quasi-submaximal spaces and related examples. More precisely, we prove that if X is a quasi-submaximal and nodec space, then X is submaximal. As properties of quasi-submaximality, we show that if X is a quasi-submaximal space, then (a) for every dense $\small{D{\subset}X}$, Int(D) is dense in X, and (b) there are no disjoint dense subsets. Also, we illustrate some basic facts and examples giving the relationships among the properties mentioned in this paper.
Keywords
maximal spaces;submaximal spaces;quasi-submaximal spaces;digital planes;digital lines;
Language
English
Cited by
References
1.
M. E. Adams, Kerim Relaid, Lobna Dridi, and Othman Echi, Submarimal and spectral spaces, Mathematical Proceedings of t he Royal Irish Academy, 108A (2) (2008), 137-147.

2.
B. Al-Nashef, On semipreopen sets, Questions and Answers in General Topology, 19 (2001), 203-212.

3.
A. V. Arhangel'skii, P. J. Collins, On submaximal spaces, Topology and its Applications, 64 (1995), 219-241.

4.
N. Bourbaki. General topology: Chapter 1-4, Translated from the French, Reprint of the 1989 English translation, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1998.

5.
J. Dontchev, On door spaces, Indian J. pure appl. Math., 26(9) (1995), 873-881.

6.
J. Dontchev, On submaximal spaces, Tamkang Journal of Mathematics 26 (1995), 243-50.

7.
M. Fusimoto, H. Makl, T. Noiri and S. Takigawa , The digital plane is quasisubmarimal, Questions and Answers in General Topology, 22 (2004), 163-168.

8.
M. Fusimoto, S. Takigawa, J. Dontchev, T. Noiri and H. Maki, The topological stucture and groups of digital n-spaces, Kochi J. Math., 1 (2006). 31-55.

9.
J.A. Guthrie, H.E. Stone, and M.L. Wage, Maximal connected Hausdorff topologies, Topology Proc., 2 (1977), 349-353.

10.
J. L. Kelley, General Topology, D. Van Nostrand Company, Inc. Princeton, new Jersey, 1955.

11.
E.D Khalimsky, R Kopperman and P.R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and Its applications 36 (1990), 1-17.

12.
T.Y. Kong, R. Kopperman and P.R. Meyer, A topological approach to digital topology, American Mathematical Monthly 98 (1991), 901-17.

13.
O. Njastad, On some classes of nearly open sets, Pacific J. Math., 15, (1965), 961-970.

14.
J. Schroder, Some answers concerning submnximal spaces, Questions and Answers in General Topology 17 (1919), 221-225.

15.
Eric K. van Douwen, Applications of maximal topologies, Topology and its Applications, 51 (2) (1993), 125-139.