In this paper, the local function, the weak semi-local function and the local
closure function are compared with each other according to the inclusion
relation. We define a new operator by using the weak semi-local function and
investigate its properties. Thanks to this operator, we obtain two new
topologies which are finer than some previously defined topologies.
<abstract><p>In this study, a $ \zeta^*_\Gamma $-local function is defined and its properties are examined. This newly defined local function is compared with the well-known local function and the local closure function according to the relation of being a subset. With the help of this new local function, the $ \Psi_{\zeta^*_\Gamma} $ operator is defined and topologies are obtained. Moreover, alternative answers are given to an open question found in the literature. $ \Psi_{\zeta^*_\Gamma} $-compatibility is defined and its properties are examined. $ \Psi_{\zeta^*_\Gamma} $-compatibility is characterized with the help of the new operator. Finally, new spaces were defined and characterized.</p></abstract>
The definitions of new type separated subsets are given in ideal topological spaces. By using these definitions, we introduce new types of connectedness. It is shown that these new types of connectedness are more general than some previously defined concepts of connectedness in ideal topological spaces. The new types of connectedness are compared with well-known connectedness in point-set topology. Then, the intermediate value theorem for ideal topological spaces is given. Also, for some special cases, it is shown that the intermediate value theorem in ideal topological spaces and the intermediate value theorem in topological spaces coincide.
It is well known that bitopologies associated with knot digraphs is finded by using knot digraph notation. In this work, we have developed a method that we called reverse of knot digraph notation to find out which knot belongs to when a bitopology associated with the knot is given.
The bitopologies have been associated with some knots in the literature with the help of a method called the knot digraph notation. The knot graphs and quasi pseudo metric spaces were used to obtain these bitopologies. With the help of quasi pseudo metrics, two topologies were obtained on a set. In this way, an association between some knots and bitopologies was established. The authors sought an answer to the question “Given the bitopologies associated with knots, can the knot itself be obtained ?” and they gave a method. This mentioned method consists of 6 steps.. In this work, it is shown in detail that according to the Alexander-Briggs notation, the reverse of the knot digraph notation is provided for the knots 3(1), 5(1), 5(2), 6(1), 6(2), 7(1), 7(2), 7(3), 8(1), 8(2), 8(3), 9(1), 9(2), 9(3), 10(1), 10(2), 10(3).
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