Most of our real life problems in engineering, social and medical science, economics, environment, etc., involve imprecise data and their solutions involve the use of mathematical principles based on uncertainty and imprecision. To handle such uncertainties, Zadeh (1965) introduced the concept of fuzzy set (FS) and fuzzy set operations. The analytical part of fuzzy set theory was practically started with the paper of Chang (1968) who introduced the concept of fuzzy topological spaces. However, this theory is associated with an inherent limitation, which is the inadequacy of the parametrization tool associated with this theory as it was mentioned by Molodtsov (1999). Molodtsov (1999) introduced the concept of the soft set (SS) theory which is free from the aforementioned problems and started to develop the basics of the corresponding theory as a new approach for modeling uncertainties. Shabir and Naz (2011) studied the topological structures of soft sets. Intuitionistic fuzzy set theory was introduced by Atanassov (1986). In recent times, the process of fuzzification of soft set theory is rapidly progressed. Maji et al. (2001a, b) combined the theory of SS with the fuzzy and intuitionistic fuzzy set theory and called as fuzzy soft set (FSS) and intuitionistic fuzzy soft set (IFSS). Topological structure of fuzzy soft sets was started by Tanay and Burc Kandemir (2011). The study was pursued by some others researchers (Chakraborty et al., 2014; Gain et al., 2013; Mukherjee et al., 2015). Majumdar and Samanta (2010) introduced generalized fuzzy soft set (GFSS) and successfully applied their notion in a decision making problem. Yang (2011) pointed out that some results of Majumdar and Samanta (2010) are not valid in general. Chakraborty and Mukherjee (2015) introduced generalized fuzzy soft union, generalized fuzzy soft intersection and severalother properties of generalized fuzzy soft sets. Also, they introduced generalized fuzzy soft topological spaces, generalized fuzzy soft closure, generalized fuzzy soft interior and studied some of their properties. Arora and Garg (2017a, b) solved the MCDM problem and established the IFSWA operator and the IFSWG operator under the IFSS environment. Garg (2017) introduced some series of averaging aggregation operators have been presented under the intuitionistic fuzzy environment. Garg and Arora (2017a, b) introduced distance and similarity measures for dual hesitant fuzzy soft sets in multi-criteria decision making problem, also they presented some generalized and group-based generalized intuitionistic fuzzy soft sets in decision-making.

Here, the basic definitions and results which will be needed in the sequel were presented.

 

 

 

 

 

 

 

 

Lemma 2

 

Similarly, one can be deduce similar diagram of the relationship between analogues topologies.

In this paper, we have introduced generalized fuzzy soft point, generalized fuzzy soft open base and subbase. The generalized fuzzy soft topological subspaces is introduced. Finally, we concluded that GFSS and IFSS are independent notions, whereas each of them is GIFSS. So, one can try to introduce some special properties of compactness, some separation axioms, connected nessetc on generalized fuzzy soft topological spaces.

In this paper, we have introduced generalized fuzzy soft point, generalized fuzzy soft open base and subbase. The generalized fuzzy soft topological subspaces is introduced. Finally, we concluded that GFSS and IFSS are independent notions, whereas each of them is GIFSS. So, one can try to introduce some special properties of compactness, some separation axioms, connected nessetc on generalized fuzzy soft topological spaces.