MINIMAL BASICALLY DISCONNECTED COVERS OF PRODUCT SPACES

In this paper, we show that if the minimal basically disconnected cover ${\wedge}X_\imath\;of\;X_\imath$ is given by the space of fixed a $Z(X)^#$-ultrafilters on $X_\imath\;(\imath=1,2)\;and\;{\wedge}X_1\;{\times}\;{\wedge}X_2$ is a basically disconnected space, then ${\wedge}X_1\;{\times}\;{\wedge}X_2$ is the minimal basically disconnected cover of $X_1\;{\times}\;X_2$. Moreover, observing that the product space of a P-space and a countably locally weakly Lindelof basically disconnected space is basically disconnected, we show that if X is a weakly Lindelof almost P-space and Y is a countably locally weakly Lindelof space, then (${\wedge}X\;{\times}\;{\wedge}Y,\;{\wedge}_X\;{\times}\;{\wedge}_Y$) is the minimal basically disconnected cover of $X\;{\times}\;Y$.