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<p>In this paper, we first prove that the retract of a consonant space (or co-consonant space) is consonant (co-consonant). Simultaneously, we consider the co-consonance of two powerspace constructions and proved that (1) the co-consonance of the Smyth powerspace <i>P</i><sub><i>S</i></sub>(<i>X</i>) implies the co-consonance of <i>X</i> if <i>X</i> is strongly compact; (2) the co-consonance of <i>X</i> implies the co-consonance of the Smyth powerspace under some conditions; (3) if the lower powerspace <i>P</i><sub><i>H</i></sub>(<i>X</i>) is co-consonant, then <i>X</i> is co-consonant; (4) for a continuous poset <i>P</i>, the lower powerspace <i>P</i><sub><i>H</i></sub>(Σ<i>P</i>) is co-consonant.</p>

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Some Problems about Consonant and Co-Consonant Spaces

  • 期刊论文

  • 发表日期：

    2025年4月15日

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  • Zhengmao He,
  • Bin Zhao

摘要

In this paper, we first prove that the retract of a consonant space (or co-consonant space) is consonant (co-consonant). Simultaneously, we consider the co-consonance of two powerspace constructions and proved that (1) the co-consonance of the Smyth powerspace PS(X) implies the co-consonance of X if X is strongly compact; (2) the co-consonance of X implies the co-consonance of the Smyth powerspace under some conditions; (3) if the lower powerspace PH(X) is co-consonant, then X is co-consonant; (4) for a continuous poset P, the lower powerspace PH(ΣP) is co-consonant.