Korean J. Math. Vol. 34 No. 2 (2026) pp.225-231
DOI: https://doi.org/10.11568/kjm.2026.34.2.225

Scatteredness of hyperspaces

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Published: Jun 30, 2026
Keywords:
Topological dimension, totally disconnectedness, scatteredness, separation property, Urysohn Lemma
Mathematics Subject Classification:
54B20, 54G12, 54D15, 54F50

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Namjip Koo
Department of Mathematics, Chungnam National University, Daejeon 34134, Korea
https://orcid.org/0000-0003-0180-9007
Hyunhee Lee
Department of Mathematics, Chungnam National University, Daejeon 34134, Korea
https://orcid.org/0000-0001-8722-2092

Abstract

In this paper, we study separation properties on the space of closed subsets of a totally disconnected compact space. Thus we show that a topological space $X$ is scattered if and only if the corresponding hyperspace $2^X$ is also scattered under the condition of local finiteness in $X$. We also provide characterizations concerning separation properties of the hyperspace via continuous real-valued functions. Furthermore, we give some examples related to our results.



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Supporting Agencies

This work was supported by Global - Learning {\&} Academic research institution for Master's · PhD students, and Postdoc(G-LAMP) Program of the National Research Foundation of Korea(NRF) grant funded by the Ministry of Education(No. RS-2025-25442707).

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