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The Answer to a Problem Posed by Zhao and Ho

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Abstract

Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ.

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Acknowledgements

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

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, P. R. China

    Bin Zhao & Jing Lu

  2. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, P. R. China

    Kai Yun Wang

Authors
  1. Bin Zhao
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  2. Jing Lu
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  3. Kai Yun Wang
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Corresponding author

Correspondence to Kai Yun Wang.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 11531009 and 11871320) and the Fundamental Research Funds for the Central Universities (Grant No. GK201803002)

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Zhao, B., Lu, J. & Wang, K.Y. The Answer to a Problem Posed by Zhao and Ho. Acta. Math. Sin.-English Ser. 35, 438–444 (2019). https://doi.org/10.1007/s10114-018-7535-6

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  • DOI: https://doi.org/10.1007/s10114-018-7535-6

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