COMPACTIFICATIONS OF A FIXED SET

Year 2021, Volume: 3 Issue: 1, 1 - 5, 29.04.2021

Ramkumar Solai

https://doi.org/10.47087/mjm.826165

Abstract

Compactification of a space X is a compact space containing X as a dense subspace. Magills construction for compactications of a fixed Tychonoff space through partitions is applied to derive compactications of various Tychonoff spaces (X;T), with a fixed set X and with a variation in Tychonoff topologies 'T'. Some possible extensions of mappings are obtained in this regard. Magills construction for compactications of a fixed Tychonoff space through partitions is applied to derive compactications of various Tychonoff spaces (X;T), with a fixed set X and with a variation in Tychonoff topologies 'T'. Some possible extensions of mappings are obtained in this regard. In a compact extension of a topological group, the inverse operation should be extendable homeomorphically from the base topological group. Finally mappings are extended homeomorphically from topological space to its compact extension, when topologies are also varied.

Keywords

Hausdor compactications , Extension of mappings , Complete upper semi lattices

References

• [1] A. V. Arhangel'skii, General topology II, Springer, Berlin, 1996.
• [2] T. Alaste, Function lattices and compactifications, Appl. Gen. Topol., 15(2), 2014, 183-202.
• [3] G. Bezhanishvili, P. J. Morandi and B. Olberding, An extension of de Vries duality to completely regular spaces and compactifications, Topology Appl., 257, (2019), 85 --105.
• [4] A. A. Chekeev and T. J. Kasymova, Ultrafilter-completeness on zero-sets of uniformly continuous functions, Topology Appl., 252, (2019), 27 - 41.
• [5] G. Gratzer, General lattice theory, Academic press, New York, (1978).
• [6] A. W. Gutierrez, On the metric compactification of infinite-dimensional Lp spaces, arXivpreprint arXiv:1802.04710, (2018).
• [7] W. He and Z. Xiao, Lattice of compactifications of a topological group, Categories and General Algebraic Structures with Applications, 10(1), (2019), 39 - 50.
• [8] A. T. Junhan, A Report on Hausdorff Compactifications of R, arXiv preprint arXiv:1901.08167, (2019).
• [9] M. Krukowski, Arzela-Ascoli theorem via the Wallman compactification, Quaestiones Mathematicae, 41(3), (2018), 349 - 357.
• [10] K.D. Magill, jr, The lattice of compactifications of a locally compact space, Proc. London. math. soc., 18, (1968), 231 - 244.
• [11] C. G. Moorthy, K. T. Rajambal, K. S. Lakshmi, R. Parvatham and H. M. Srivastava, A general compactification for Hausdorff spaces, Proceedings of the international conference on Analysis and its applications, (eds. KS Lakshmi, R. Parvatham, and HM Srivastava, Allied publishers Ltd), (2001), 67 - 71.
• [12] J. R. Porter and R. Grandwoods, Extensions and absolutes of Hausdorff spaces, Springer-Verlag, Newyork, (1988).
• [13] S. Ramkumar and C. G. Moorthy, A construction for topological extensions, Asian-Eur. J. Math., 4(3), (2011), 481 - 494.
• [14] S. Ramkumar and C. G. Moorthy, A construction for semi-compactifications, Asian-Eur. J. Math., 5(1), (2012), 1250008.
• [15] S. Ramkumar and C. Ganesa Moorthy, Extendability of semi-metrics to compactifications, Asian-Eur. J. Math., 6(1), (2013), 1350015.