×

Mappings and decompositions of continuity on almost Lindelöf spaces. (English) Zbl 1117.54024

Summary: A topological space \(X\) is said to be almost Lindelöf if for every open cover \(\{U_{\alpha }:\alpha \in \Delta \}\) of \(X\) there exists a countable subset \(\{\alpha _{n}:n\in \mathbb N\}\subseteq \Delta \) such that \(X=\cup _{n\in \mathbb N}\text{Cl}(U_{\alpha _{n}})\). In this paper we study the effect of mappings and some decompositions of continuity on almost Lindelöf spaces. The main result is that a \(\theta\)-continuous image of an almost Lindelöf space is almost Lindelöf.

MSC:

54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54C08 Weak and generalized continuity
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Abd El-Monsef, M. E.; El-Deeb, S. N.; Mahmoud, R. A., <mml:math alttext=“\( \beta \)” id=“C1”>β-open sets and <mml:math alttext=“\( \beta \)” id=“C2”>β-continuous mapping, Bulletin of the Faculty of Science. Assiut University. A. Physics and Mathematics, 12, 1, 77-90 (1983) · Zbl 0577.54008
[2] Baker, C. W., On super continuous functions, Bulletin of the Korean Mathematical Society, 22, 1, 17-22 (1985) · Zbl 0557.54009
[3] Baker, C. W., Subcontra-continuous functions, International Journal of Mathematics and Mathematical Sciences, 21, 1, 19-24 (1998) · Zbl 0888.54017 · doi:10.1155/S0161171298000027
[4] Balasubramanian, G., On some generalizations of compact spaces, Glasnik Matematički. Serija III, 17(37), 2, 367-380 (1982) · Zbl 0514.54015
[5] Cammaroto, F.; Santoro, G., Some counterexamples and properties on generalizations of Lindelöf spaces, International Journal of Mathematics and Mathematical Sciences, 19, 4, 737-746 (1996) · Zbl 0860.54033 · doi:10.1155/S0161171296001020
[6] Dontchev, J., Contra-continuous functions and strongly -closed spaces, International Journal of Mathematics and Mathematical Sciences, 19, 2, 303-310 (1996) · Zbl 0840.54015 · doi:10.1155/S0161171296000427
[7] Dontchev, J.; Przemski, M., On the various decompositions of continuous and some weakly continuous functions, Acta Mathematica Hungarica, 71, 1-2, 109-120 (1996) · Zbl 0852.54012 · doi:10.1007/BF00052199
[8] Fawakhreh, A. J.; Kılıçman, A., Mappings and some decompositions of continuity on nearly Lindelöf spaces, Acta Mathematica Hungarica, 97, 3, 199-206 (2002) · Zbl 1017.54008 · doi:10.1023/A:1020803027828
[9] Fomin, S. V., Extensions of topological spaces, Annals of Mathematics. Second Series, 44, 471-480 (1943) · Zbl 0061.39601
[10] Kim, H. O., Notes on -compact spaces and functionally compact spaces, Kyungpook Mathematical Journal, 10, 75-80 (1970) · Zbl 0205.27001
[11] Long, P. E.; Herrington, L. L., Strongly <mml:math alttext=“\( \theta \)” id=“C5”>θ-continuous functions, Journal of the Korean Mathematical Society, 18, 1, 21-28 (1981) · Zbl 0478.54006
[12] Mashhour, A. S.; Abd El-Monsef, M. E.; El-Deep, S. N., On precontinuous and weak precontinuous mappings, Proceedings of the Mathematical and Physical Society of Egypt, 53, 47-53 (1983) (1982) · Zbl 0571.54011
[13] Nasef, A. A.; Noiri, T., Some weak forms of almost continuity, Acta Mathematica Hungarica, 74, 3, 211-219 (1997) · Zbl 0924.54017 · doi:10.1023/A:1006507816942
[14] Noiri, T., On <mml:math alttext=“\( \delta \)” id=“C6”>δ-continuous functions, Journal of the Korean Mathematical Society, 16, 2, 161-166 (1980) · Zbl 0435.54010
[15] Park, J. H.; Ha, H. Y., A note on weakly quasi-continuous functions, International Journal of Mathematics and Mathematical Sciences, 19, 4, 767-772 (1996) · Zbl 0863.54011 · doi:10.1155/S0161171296001068
[16] Popa, V.; Stan, C., On a decomposition of quasi-continuity in topological spaces, Studii şi Cercetări Matematice, 25, 41-43 (1973) · Zbl 0255.54008
[17] Singal, M. K.; Prabha Arya, S., On almost-regular spaces, Glasnik Matematički. Serija III, 4 (24), 89-99 (1969) · Zbl 0169.24902
[18] Singal, M. K.; Prabha Arya, S., On nearly paracompact spaces, Matematički Vesnik, 6 (21), 3-16 (1969) · Zbl 0177.50802
[19] Singal, M. K.; Singal, A. R., Almost-continuous mappings, Yokohama Mathematical Journal, 16, 63-73 (1968) · Zbl 0191.20802
[20] Singal, M. K.; Singal, A. R., Mildly normal spaces, Kyungpook Mathematical Journal, 13, 27-31 (1973) · Zbl 0266.54006
[21] Willard, S.; Dissanayake, U. N. B., The almost Lindelöf degree, Canadian Mathematical Bulletin, 27, 4, 452-455 (1984) · Zbl 0551.54003
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.