Received:April 14, 2017  Revised:August 05, 2017
Key Word: topological spaces   $G$-methods   pointwise $G$-methods   $G$-hull-closed sets  ~$G$-kernel-open sets  ~$G$-kernel-neighborhoods  ~$G$-kernel-derived sets.  
Fund ProjectL:This research is supported by Ningde Normal University (Nos. 2016Q34) and NSFC (Nos. 11471153).,The Key Program of the Chinese Academy of Sciences,The Major Program of the Chinese Academy of Sciences,The Knowledge Innovation Program of the Chinese Academy of Sciences,100 Talents Program of the Chinese Academy of Sciences,China Postdoctoral Science Foundation,Joint Research Fund for Hong Kong and Macao Young Scholars,市自然科学基金
Author NameAffiliationE-mail
liu li Ningde Normal University 429378220@qq.com 
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      Based on $G$-hulls and $G$-kernels under the meaning of $G$-methods on sets, we introduce the concepts of ~$G$-hull-closed sets,~$G$-kernel-open sets,~$G$-kernel-neighborhoods and ~$G$-kernel-derived sets, discuss some related properties. In particular, we define pointwise $G$-methods, prove the consistency of $G$-closed sets and $G$-hull-closed sets,~$G$-open sets and $G$-kernel-open sets,~$G$-neighborhoods and $G$-kernel-neighborhoods,~$G$-derived sets and~$G$-kernel-derived sets under this method, and enrich some results about $G$-closed sets, $G$-open sets, $G$-interiors,~$G$-neighborhoods and $G$-derived sets in sets. At the same time, we put forward some problems for further research.
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