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Hereafter, the concept of a function weighted quasi-metric space is introduced. A necessary topology on this new structure is considered. A condition that ensures the uniqueness of the limit of a sequence in such a space is provided. A relation between the bi-convergent sequences and the bi-Cauchy ones are proved. Certain classes of operators with respect to their fixed point properties are investigated, having in view this framework. Examples that support our results are also included.