A compact and sober topological space \(X\) is called spectral if the family of compact open sets of \(X\) is closed under finite intersections and forms a basis of \(X.\) The author obtains some partial results concerning the following question: For which \(T_0\)-spaces \(X\) is a construction due to Herrlich, which is called the \(T_0\)-compactification of \(X\), spectral? In this context those \(T_1\)-spaces \(X\) are characterized whose Wallman compactification is spectral. Reviewer's remark: The condition provided describes the normal strongly zero-dimensional spaces.