It is shown that every continuous image of a compact Hausdorff α–scattered space X(i.e., every subset A of X with |A|≥α has an isolated point relative to A and α is the least regular cardinal with this property) is b –scattered for some b≤α. Consequently, if X is compact Hausdorff α –scattered where α≤c and c is the cardinally of continuum, then a=N_0 the first infinite cardinal and X is scattered. Surprisingly, it follows that in any compact Hausdorff space X, every non-empty subset has an isolated point if and only if every subset of X has an isolated point.
