The author shows that for every completely regular Hausdorff space X there exists the smallest basically disconnected space

${\Lambda}$ X which has a canonical perfect irreducible mapping onto X; i.e. there exists a perfect irreducible mapping

${\Lambda}$ :

${\Lambda}$ $X{\to}^{onto}X$ such that for every perfect irreducible mapping

$g:Y{\to}^{onto}X$, where Y is basically disconnected, there exists a continuous mapping

$h:Y{\to}^{onto}{\Lambda}X$ such that

$g={\Lambda}\circ h$. In the first stage of the construction the author proves that the space

${{\Lambda}}_{1}X$ consisting of all prime prime-z-filters which are generated by open ultrafilters is homeomorphic to X iff X is basically disconnected. Next the space

${\Lambda}$ X is constructed as an inverse limit of a continuous inverse sequence

$\{{{\Lambda}}_{\alpha}X,{{\Lambda}}_{\beta}^{\alpha}$;

$\beta <\alpha <{\omega}_{1}\}$, where

${{\Lambda}}_{\alpha +1}X={{\Lambda}}_{1}\left({{\Lambda}}_{\alpha}X\right)$ for every

$\alpha <{\omega}_{1}$. From the existence of the space

${\Lambda}$ X it follows that for every locally compact basically disconnected space X there exists the smallest basically disconnected compactification BX.