## Abstract

Let (*U*, *R*) be an approximation space with *U* being non-empty set and *R* being an equivalence relation on *U*, and let \({\overline{G}}\) and \({\underline{G}}\) be the upper approximation and the lower approximation of subset *G* of *U*. A topological rough group *G* is a rough group \(G=({\underline{G}}, {\overline{G}})\) endowed with a topology, which is induced from the upper approximation space \({\overline{G}}\), such that the product mapping \(f: G\times G\rightarrow {\overline{G}}\) and the inverse mapping are continuous. In the class of topological rough groups, the relations of some separation axioms are obtained; some basic properties of the neighborhoods of the rough identity element and topological rough subgroups are investigated. In particular, some examples of topological rough groups are provided to clarify some facts about topological rough groups. Moreover, the version of open mapping theorem in the class of topological rough group is obtained. Further, some interesting open questions are posed.

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## Acknowledgements

The first author is supported by the Key Program of the Natural Science Foundation of Fujian Province (No: 2020J02043), the NSFC (No. 11571158), the Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics. The second author is supported by the Young and middle-aged project in Fujian Province (No. JAT190397). The fourth author is supported by the NSFC (No. 11871259).

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*This paper is dedicated to professor Shou Lin on the occasion of his 60th birthday.*

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Lin, F., Sun, Q., Lin, Y. *et al.* Some topological properties of topological rough groups.
*Soft Comput* (2021). https://doi.org/10.1007/s00500-021-05631-6

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### Keywords

- Rough group
- Topological rough group
- Topological rough subgroup
- Strongly topological rough group
- Connected; extremally disconnected
- Separation axiom
- Rough homomorphism
- Rough kernel
- Topological group
- Lower approximation
- Upper approximation