# zbMATH — the first resource for mathematics

Characterizations of strongly compact spaces. (English) Zbl 1179.54037
Summary: A topological space $$(X,\tau)$$ is said to be strongly compact if every preopen cover of $$(X,\tau)$$ admits a finite subcover. In this paper, we introduce a new class of sets called $${\mathcal N}$$-preopen sets which is weaker than both open sets and $${\mathcal N}$$-open sets. Where a subset $$A$$ is said to be $${\mathcal N}$$-preopen if for each $$x\in A$$ there exists a preopen set $${\mathcal U}_x$$ containing $$x$$ such that $${\mathcal U}_x - A$$ is a finite set. We investigate some properties of the sets. Moreover, we obtain new characterizations and preserving theorems of strongly compact spaces.

##### MSC:
 54D30 Compactness
##### Keywords:
strongly compact; $${\mathcal N}$$-preopen sets
Full Text:
##### References:
 [1] R. H. Atia, S. N. El-Deeb, and I. A. Hasanein, “A note on strong compactness and S-closedness,” Matemati\vcki Vesnik, vol. 6, no. 1, pp. 23-28, 1982. · Zbl 0501.54015 [2] A. S. Mashhour, M. E. Abd El-Monsef, I. A. Hasanein, and T. Noiri, “Strongly compact spaces,” Delta Journal of Science, vol. 8, no. 1, pp. 30-46, 1984. [3] M. Ganster, “Some remarks on strongly compact spaces and semi-compact spaces,” Bulletin Malaysian Mathematical Society, vol. 10, no. 2, pp. 67-70, 1987. · Zbl 0668.54016 [4] D. S. Janković, I. L. Reilly, and M. K. Vamanamurthy, “On strongly compact topological spaces,” Questions and Answers in General Topology, vol. 6, no. 1, pp. 29-40, 1988. · Zbl 0647.54018 [5] S. Jafari and T. Noiri, “Strongly compact spaces and firmly precontinuous functions,” Research Reports. Yatsushiro National College of Technology, vol. 24, pp. 97-100, 2002. [6] S. Jafari and T. Noiri, “More on strongly compact spaces,” Missouri Journal of Mathematical Sciences, vol. 19, no. 1, pp. 52-61, 2007. · Zbl 1138.54023 [7] A. S. Mashhour, M. E. Abd El-Monsef, and S. N. El-Deeb, “On precontinuous and weak precontinuous mappings,” Proceedings of the Mathematical and Physical Society of Egypt, vol. 53, pp. 47-53, 1982. · Zbl 0571.54011 [8] A. Al-Omari and M. S. M. Noorani, “New characterizations of compact spaces,” submited. · Zbl 1207.54033 [9] I. L. Reilly and M. K. Vamanamurthy, “On some questions concerning preopen sets,” Kyungpook Mathematical Journal, vol. 30, no. 1, pp. 87-93, 1990. · Zbl 0718.54004 [10] O. Njåstad,, “On some classes of nearly open sets,” Pacific Journal of Mathematics, vol. 15, pp. 961-970, 1965. · Zbl 0137.41903 [11] V. Popa and T. Noiri, “Almost weakly continuous functions,” Demonstratio Mathematica, vol. 25, no. 1-2, pp. 241-251, 1992. · Zbl 0789.54014 [12] A. S. Mashhour, I. A. Hasanein, and S. N. El-Deeb, “A note on semicontinuity and precontinuity,” Indian Journal of Pure and Applied Mathematics, vol. 13, no. 10, pp. 1119-1123, 1982. · Zbl 0499.54009 [13] A. S. Mashhour, M. E. Abd El-Monsef, and I. A. Hasanein, “On pretopological spaces,” Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, vol. 28, no. 1, pp. 39-45, 1984. · Zbl 0532.54002 [14] I. L. Reilly and M. K. Vamanamurthy, “On \alpha -continuity in topological spaces,” Acta Mathematica Hungarica, vol. 45, no. 1-2, pp. 27-32, 1985. · Zbl 0576.54014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.