The principle point ofithis paper is to present new type of topologicalispaces are introduced by nano topology and so on Nano topology introduced by Thivagar using this nano topology we introduced micro topology and also introduce new type of open sets namely of Micro-α-open sets furthermore, a portion of their properties are researched. As application to Micro-α-open sets we introduce Micro-α-continuous functions also, get a portion of their essential properties

In this paper we introduces a new definition , called i- open and via this definition we  introduce class of topological concepts(µ-θ-i-open set, µ-θ-i-closed, strong faintly µ-θ-continuity,  strong µ-θ-continuity )and we generalized these concepts in bi -supra topological space .At last  many  important theorems in strongly faintly M-θ-i-continuous functions are investigated. And study  the relationships among  these functions and other forms are discussed.

Every year different types of topological spaces are introduced by many topologists. Nowadays available topologies are supra topology, ideal topology, bitopology, fuzzy topology, Fine topology, nano topology and so on. Nano topology introduced by Thivagar, using this nano topology we introduced micro topology and also study the concepts of θ - micro open sets and θ – micro continuous and some of their properties are investigated.

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In the presentipaper, authors establish some resultsiinvolving in coefficient bounds, extremeipoints, convex combination, Here we alsoidetermine that the class studiediin this paper is closediunder convolution finally, introduce and study partial sumsiof the Liberaiintegral operator and inclusion relationship involving neighborhoods for a new subclass ofiharmonic univalent functionsidefined by fractional calculusioperator in theiunit disk U = { z ∈ ℂ : | z | < 1 } .