Groups of \(\theta\)-generalized homeomorphisms and the digital line (Q1296305)

The \(\theta\)-closure of a set \(A\subset (X,\tau)\) is defined as cl\(_\theta(A) =\{x\in X:\text{ cl}(U)\cap A\neq\emptyset, U\in\tau,x\in U\}\) and a set \(C\subset X\) is said to be \(\theta\)-generalized closed if cl\(_\theta(C)\subset U\) whenever \(C\subset U\) and \(U\in \tau\). A \(\theta\)-gc-homeomorphism is a bijective function \(f\) such that both it and its inverse preserve \(\theta\)-generalized closed sets. The authors study these and a number of other related types of mappings. Among the results, it is shown that the set of \(\theta\)-gc-homeomorphisms on a space \(X\) has a group structure and a characterization of the \(\theta\)-generalized closed subsets of the Khalimsky line is given.