In this study, a -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 operator is defined and topologies are obtained. Moreover, alternative answers are given to an open question found in the literature. -compatibility is defined and its properties are examined. -compatibility is characterized with the help of the new operator. Finally, new spaces were defined and characterized.