Abstract

In this Paper, we will show that every basically disconnected space is a projective object in the category <TEX>$Tych_{\sigma}$</TEX> of Tychonoff spaces and <TEX>$_{\sigma}Z^{#}$</TEX> -irreducible maps and that if X is a space such that <TEX>${\Beta} {\Lambda} X={\Lambda} {\Beta} X$</TEX>, then X has a projective cover in <TEX>$Tych_{\sigma}$</TEX>. Moreover, observing that for any weakly Linde1of space, <TEX>${\Lambda} X : {\Lambda} X\;{\longrightarrow}\;X$</TEX> is <TEX>$_{\sigma}Z^{#}$</TEX>-irreducible, we will show that the projective objects in <TEX>$wLind_{\sigma}$</TEX>/ of weakly Lindelof spaces and <TEX>$_{\sigma}Z^{#}$</TEX>-irreducible maps are precisely the basically disconnected spaces.