Abstract
Let \({\mathfrak{I}}^{{\varvec{C}}{\varvec{L}}}\boldsymbol{ }({\varvec{n}})\) be the number of clopen topologies on a finite set of \({\varvec{n}}\) elements. It is proved that the explicit formula for finding the total number of clopen topologies is\({\sum }_{{\varvec{k}}=1}^{{\varvec{n}}}\mathcal{S}\left({\varvec{n}},{\varvec{k}}\right)={\sum }_{{\varvec{i}}=1}^{{\varvec{n}}}{\text{CL}}({\varvec{n}},{2}^{{\varvec{i}}})\), where \(\mathcal{S}\left({\varvec{n}},{\varvec{k}}\right)\) is the Stirling number of the second kind and \({\text{CL}}({\varvec{n}},{2}^{{\varvec{i}}})\) is the number of clopen topologies having \({2}^{{\varvec{i}}}\) open sets, \({\varvec{i}}\boldsymbol{ }=\boldsymbol{ }1,\boldsymbol{ }2,\boldsymbol{ }3,\boldsymbol{ }.\boldsymbol{ }.\boldsymbol{ }.\boldsymbol{ },\boldsymbol{ }{\varvec{n}}\). Some results concerning the number of clopen topological spaces whose topologies have the same cardinality are also obtained.
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Basumatary, B., Basumatary, J. & Nordo, G. On the number of clopen topological spaces on a finite set. Afr. Mat. 34, 57 (2023). https://doi.org/10.1007/s13370-023-01091-3
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DOI: https://doi.org/10.1007/s13370-023-01091-3