On \(\delta\)-continuous multifunctions and paralindelöfness (Q1911825)

The authors extend the notion of a \(\delta\)-continuous map introduced by \textit{T. Noiri} [J. Korean Math. Soc. 16, 161-166 (1980; Zbl 0435.54010)] to multifunctions, i.e., maps which carry a point to a subset, and prove theorems concerning the invariance of some covering properties under multifunctions. A multifunction \(F:X \to Y\) is called upper (resp. lower) \(\delta\)-continuous at \(x\in X\) if for each open subset \(V\) of \(Y\) with \(F(x) \subseteq V\) (resp. \(F(x) \cap V\neq \emptyset)\), there is an open neighborhood \(U\) of \(x\) such that \(F(z) \subset \text{int(cl} V))\) (resp. \(F(z)\cap \text{int(cl} V)\neq\emptyset)\) for each \(z\in \text{int(cl} U)\). A multifunction is called \(\delta\)-continuous if it is upper and lower \(\delta\)-continuous at each \(x\in X\). A subset \(A\) of \(X\) is called an \(NPL\)-set in \(X\) if every cover of \(A\) by open sets in \(X\) has a locally countable open refinement \(\Omega\) in \(X\) with \(A\subset\bigcup \{\text{int(cl} V): V\in\Omega\}\), and a subset \(B\) of \(Y\) is called a \(C_1\)-closed set in \(Y\) if every cover of \(B\) by regular open sets in \(Y\) has a countable subcover. One of the main results is: Let \(F:X\to Y\) be a \(\delta\)-closed, \(\delta\)-continuous multifunction of a \(P\)-space \(X\) onto a \(P\)-space \(Y\) such that \(F(x)\) is \(C_1\)-closed in \(Y\) for each \(x\in X\) and \(F^-(y)\) is an \(NPL\)-set in \(X\) for each \(y\in Y\). Then, if \(K\) is an \(NPL\)-set in \(Y\), then \(F^-(K)\) is an \(NPL\)-set in \(X\). Here, \(F^-(K)= \{x\in X: F(x)\cap K \neq\emptyset\}\) and \(F^-(y)=F^-(\{y\})\).