New subspaces of extensions, connectedness and local connectedness (Q1090958)

All spaces are assumed regular. Let Y be an extension of X; for \(A\subseteq X\), let \(0_ Y(A)\) (or simply 0(A)) denote \(Y-Cl_ Y(X-A)\). For \(\kappa\geq \omega\), \(A\subseteq Y\) is a \(\kappa\)-connected set of X in Y if for each cellular family \(\{U_{\alpha}: \alpha \in \Lambda \})\) of open sets in X with \(| \Lambda | <\kappa\). \(A\subseteq 0\) \((\cup \{U_{\alpha}:\alpha\in \Lambda \})\) implies \(A\subseteq 0\) \((U_{\alpha})\) for some \(\alpha\in \Lambda\). After \textit{B. Banaschewski} [Can. J. Math. 8, 395-398 (1956; Zbl 0072.177)], a point \(p\in Y\) is a \(\kappa\)-connected point of X is Y if \(\{\) \(p\}\) is a \(\kappa\)-connected set of X in Y. Let E(Y,X,\(\kappa)\) Y be the set of all \(\kappa\)-connected points of X in Y, \(E(Y,X)=\cap \{E(Y,X,\kappa):\kappa\geq \omega \}\). For X completely regular, it is shown that \(E(\beta X,X,\omega)=\beta X\), \(E(\beta X,X,\omega_ 1)=\upsilon X\) (the Hewitt realcompactification of X), and \(E(\beta X,X)=\mu X\). Furthermore, it is shown that X is connected iff E(Y,X,\(\omega)\) is, and that X is locally connected iff the set at which E(Y,X,\(\omega)\) is locally connected is precisely E(Y,X). Finally, X is pseudocompact iff \(E(Y,X,\omega)=E(Y,X)\) for each extension Y of X.