## Abstract

This paper presents a novel framework for the study of hesitant fuzzy rough sets by integrating rough sets with hesitant fuzzy sets. Lower and upper approximations of hesitant fuzzy sets with respect to a hesitant fuzzy approximation space are first defined. Properties of hesitant fuzzy approximation operators are examined. Relationships between hesitant fuzzy approximation spaces and hesitant fuzzy topological spaces are then established. It is proved that the set of all lower approximation sets based on a hesitant fuzzy reflexive and transitive approximation space forms a hesitant fuzzy topology. And conversely, for a hesitant fuzzy rough topological space, there exists a hesitant fuzzy reflexive and transitive approximation space such that the topology in the hesitant fuzzy rough topological space is exactly the set of all lower approximation sets in the hesitant fuzzy reflexive and transitive approximation space. That is to say, there exists a one-to-one correspondence between the set of all hesitant fuzzy reflexive and transitive approximation spaces and the set of all hesitant fuzzy rough topological spaces.

## Keywords

Approximation operators Hesitant fuzzy rough sets Hesitant fuzzy sets Hesitant fuzzy topological spaces Rough sets## Notes

### Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

### Funding

This study is funded by the National Natural Science Foundation of China (Nos. 11461082, 11601474), the Natural Science Foundation of Gansu Province (No. 17JR5RA284), the Research Project Funds for Higher Education Institutions of Gansu Province (No. 2016B-005), the Fundamental Research Funds for the Central Universities of Northwest MinZu University (No. 31920190055) and the first-class discipline program of Northwest Minzu University.

### Compliance with ethical standards

### Conflict of interest

The author declares that there is no conflict of interests.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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