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In the present paper, we will propose the novel notions (e.g., Q p -closed set, Q p -open set, Q p -continuous mapping, Q p -open mapping, and Q p -closed mapping) in topological spaces. Then, we will discuss the basic properties of the above notions in detail. The category of all Q p -closed (resp. Q p -open) sets is strictly between the class of all preclosed (resp. preopen) sets and g p -closed (resp. g p -open) sets. Also, the category of all Q p -continuity (resp. Q p -open ( Q p -closed) mappings) is strictly among the class of all precontinuity (resp., preopen (preclosed) mappings) and g p -continuity (resp. g p -open ( g p -closed) mappings). Furthermore, we will present the notions of Q p -closure of a set and Q p -interior of a set and explain some of their fundamental basic properties. Several relations are equivalent between two different topological spaces. The novel two separation axioms (i.e., Q p - ℝ 0 and Q p - ℝ 1 ) based on the notion of Q p -open set and Q p -closure are investigated. The space of Q p - ℝ 0 (resp., Q p - ℝ 1 ) is strictly between the spaces of pre- ℝ 0 (resp., pre- ℝ 1 ) and g p - ℝ o (resp., g p - ℝ 1 ). Finally, some relations and properties of Q p - ℝ 0 and Q p - ℝ 1 spaces are explained.

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