On a weak form of almost weakly continuous functions (Q2708144)

A subset \(S\) of a topological space \((X,\tau)\) is said to be pre-open if \(S\subset \text{Int}(\text{Cl}(S))\). \(S\) is pre-closed if its complement is pre-open. The pre-closure of \(S\), denoted by \(\text{pCl}(S)\), is the intersection of all pre-closed sets containing \(S\). A function \(f:(X,\tau)\to (Y,\sigma)\) is said to be almost weakly continuous if \(f^{-1}(V)\subset \text{Int}(\text{Cl}(f^{-1}(\text{Cl}(V))))\) for every open subset \(V\) of \(Y\) [\textit{D. S. Janković}, Int. J. Math. Math. Sci. 8, 615-619 (1985; Zbl 0577.54012)]. It is proved in [\textit{V. Popa} and \textit{T. Noiri}, Demonstr. Math. 25, No. 1/2, 241-251 (1992; Zbl 0789.54014)] that a function \(f:(X,\tau)\to (Y,\sigma)\) is almost weakly continuous if and only if \(\text{pCl}(f^{-1}(V))\subset f^{-1}(\text{Cl}(V))\) for every open subset \(V\) of \(Y\). A function \(f: (X,\tau)\to (Y,\sigma)\) is said to be subweakly continuous if there is an open base \({\mathcal B}\) for the topology on \(Y\) such that \(\text{Cl}(f^{-1}(V))\subset f^{-1}(\text{Cl}(V))\) for every \(V\in{\mathcal B}\).NEWLINENEWLINENEWLINEIn this paper a weak form of almost weak continuity, called subalmost weak continuity, is introduced. A function \(f:(X,\tau)\to (Y,\sigma)\) is said to be subalmost weakly continuous if there is an open base \({\mathcal B}\) for the topology on \(Y\) such that \(\text{pCl}(f^{-1}(V))\subset f^{-1}(\text{Cl}(V))\) for every \(B\in{\mathcal B}\). Subalmost weak continuity is strictly weaker than both almost weak continuity and subweak continuity. Subalmost weak continuity is used to improve a result of Popa and Noiri concerning the graph on almost weakly continuous function. In section 5 additional properties of subweakly continuous functions are investigated.