Some localized separation axioms and their implications |
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Authors: | K K Dube D N Misra |
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Institution: | (1) Department of Mathematics and Statistics, University of Saugar, Gour Nagar, Sagar, M. P., India |
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Abstract: | In a topological spaceX, a T2-distinct pointx means that for anyyX xy, there exist disjoint open neighbourhoods ofx andy. Similarly, T0-distinct points and T1distinct points are defined. In a Ti-distinct point-setA, we assume that eachxA is a T
i
-distinct point (i=0, 1, 2). In the present paper some implications of these notions which localize the T
i
-separation axioms (i=0, 1, 2) requirement, are studied. Suitable variants of regularity and normality in terms of T2-distinct points are shown hold in a paracompact space (without the assumption of any separation axioms). Later T0-distinct points are used to give two characterizations of the R
D
-axiom.1 In the end, some simple results are presented including a condition under which an almost compact set is closed and a result regarding two continuous functions from a topological space into a Hausdorff space is sharpened. A result which relates a limit pointv to an -limit point is stated. |
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Keywords: | Primary 54D99 Secondary 54D10 54D15 |
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