Structure of the exponent of a discrete space
Let X be a topological space and \({\mathcal A}\), \({\mathcal B}\) families of subsets of X. For \(A\in {\mathcal A}\), \(B\in {\mathcal B}\) put \([A,B]=\{(F)\in \exp X:A\subset F\subset X-B\},\) where exp X denotes the system of all closed sets of X. If \({\mathcal A}\) and \({\mathcal B}\) are invariant with respect to finite unions, then the system \(\sigma =\{[A,B]:A\in {\mathcal A},\quad B\in {\mathcal B}\}\) forms a basis of ''(\({\mathcal A},{\mathcal B})\)-topology'' that is Hausdorff topology if and only if \(A\cap B\) contains all finite subsets of X. For infinite cardinals \(\lambda_ 1\), \(\lambda_ 2\) the \((\lambda_ 1,\lambda_ 2)\)-topology on exp X is the (\({\mathcal A},{\mathcal B})\)-topology, where \({\mathcal A}=\{C\subset X:| C| <\lambda_ 1\}\) and \({\mathcal B}=\{C\subset X:| C| <\lambda_ 2\}.\) This paper deals with these \((\lambda_ 1,\lambda_ 2)\)-topologies. The following theorem (Theorem 4) is typical: ''Let X be a discrete infinite topological space, \(\lambda_ 1\), \(\lambda_ 2\) infinite cardinals, \(\lambda_ 2>| X|\). Then the space exp X with the \((\lambda_ 1,\lambda_ 2)\)-topology is homeomorphic with the direct topological sum of its copies of the number \(| X|.''\)
scientific article; zbMATH DE number 3907327