Submaximal and door compactifications |
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Authors: | Karim Belaid |
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Institution: | a University of Dammam, Faculty of Sciences of Dammam, Girls College, Department of Mathematics, PO Box 383, Dammam 31113, Saudi Arabia b University of Monastir, Higher Institute of Mathematics and Informatics (ISIM), Department of Mathematics, PO Box 223, 5000 Monastir, Tunisia c King Fahd University of Petroleum and Minerals, Department of Mathematics and Statistics, PO Box 5046, Dhahran 31261, Saudi Arabia |
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Abstract: | In this paper, a characterization is given for compact door spaces. We, also, deal with spaces X such that a compactification K(X) of X is submaximal or door.Let X be a topological space and K(X) be a compactification of X.We prove, here, that K(X) is submaximal if and only if for each dense subset D of X, the following properties hold:- (i)
- D is co-finite in K(X);
- (ii)
- for each x∈K(X)?D, {x} is closed.
If X is a noncompact space, then we show that K(X) is a door space if and only if X is a discrete space and K(X) is the one-point compactification of X. |
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Keywords: | primary 06B30 06F30 secondary 54F05 |
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