ARRANGEMENT OF ELEMENTS OF LOCALLY FINITE TOPOLOGICAL SPACES UP TO AN ALF-HOMEOMORPHISM

Han, Sang-Eon;Chun, Woo-Jik

• Received : 2011.10.11
• Accepted : 2011.12.01
• Published : 2011.12.25
• 29 2

Abstract

In relation to the classification of finite topological spaces the paper [17] studied various properties of finite topological spaces. Indeed, the study of future internet system can be very related to that of locally finite topological spaces with some order structures such as preorder, partial order, pretopology, Alexandroff topological structure and so forth. The paper generalizes the results from [17] so that the paper can enlarge topological and homotopic properties suggested in the category of finite topological spaces into those in the category of locally finite topological spaces including ALF spaces.

Keywords

Locally finite topological space;ALF space;homeomorphism;digital topology;Khalimsky topology;network

References

1. P. Alexandorff, Diskrete Rume, Mat. Sb. 2 (1937) 501-518.
2. G. Birkhoff, Lattice Theory, American Mathematical Society, 1961.
3. W. Dunham, $T{\frac{1}{2}$ spaces, Kyungpook Math. J. 17 (1977) 161-169.
4. S.E. Han, Strong k-deformation retract and its applications, Journal of Korean Mathematical Society 44(6)(2007) 1479-1503. https://doi.org/10.4134/JKMS.2007.44.6.1479
5. S.E. Han, Continuities and homeomorphisms in computer topology, Journal of Korean Mathematical Society 45(4)(2008) 923-952. https://doi.org/10.4134/JKMS.2008.45.4.923
6. S.E. Han, Equivalent ($k_0;\;k_1$)-covering and generalized digital lifting, Information Sciences 178(2)(2008) 550-561. https://doi.org/10.1016/j.ins.2007.02.004
7. S.E. Han, The k-homotopic thinning and a torus-like digital image in $\mathbf{Z}^n$, Journal of Mathematical Imaging and Vision 31 (1)(2008) 1-16. https://doi.org/10.1007/s10851-007-0061-2
8. S.E. Han, Extension problem of several continuities in computer topology, Bul- letin of Korean Mathematical Society, 47(5)(2010) 915-932. https://doi.org/10.4134/BKMS.2010.47.5.915
9. S.E. Han, Continuity of maps between axiomatic locally finite spaces and its ap- plications, International Journal of Computer Mathematics 88(14) (2011) 2889- 2900. https://doi.org/10.1080/00207160.2011.577892
10. E. Khalimsky, R. Kopperman, P.R. Meyer, Computer graphics and connected topologies on finite ordered sets, Topology and its Applications 36(1)(1991) 1-17.
11. V. Kovalevsky, Axiomatic Digital Topology, Journal of Mathematical Imaging and Vision 26 (2006) 41-58. https://doi.org/10.1007/s10851-006-7453-6
12. V. Kovalevsky, Geometry of Locally Finite Spaces, Monograph, Berlin (2008).
13. V. Kovalevsky, Sang-Eon Han, Product and hereditary property of space set topological structure, Transactions of AMS, submitted.
14. A. Rosenfeld, Connectivity in digital pictures, Journal of the ACM 17 (1970) 146-160. https://doi.org/10.1145/321556.321570
15. H. Seifert and W. Threlfall, A Textbook of Topology, Academic Press, 1980.
16. J. Stillwell, Classical Topology and Combinatorial Group Theory, Springer (1995).
17. R.E.Stong, Finite topological spaces, Transactions of AMS 123 (1966) 325-340. https://doi.org/10.1090/S0002-9947-1966-0195042-2

Acknowledgement

Grant : Development of Future Internet common conceptual architecture and inter-operation techniques for trustworthy communication

Supported by : KCA