θω−Connectedness and ω−R1 properties
AbstractWe use the theta omega closure operator to define theta omega connectedness as a property which is weaker than connectedness and stronger than θ-connectedness. We give several sufficient conditions for the equivalence between θω-connectedness and connectedness, and between θω-connectedness and θ-connectedness. We give two results regarding the union of θω-connected sets and also we show that the weakly θω-continuous image of a connected set is θω-connected. We define and investigate V -θω-connectedness as a strong form of V - θ-connectedness, and we show that the θω-connectedness and V -θω-connectedness are independent. We continue the study of R1 as a known topological property by giving several results regarding it. We introduce ω-R1 (I), ω-R1 (II), ω-R1 (III) and weakly ω-R1 as four weaker forms of R1 by utilizing ω-open sets, we give several relationships regarding them and we raise two open questions.
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