A space \(X\) is called almost resolvable iff \(X\) is the union of countably many subspaces having empty interior. If a space is not almost resolvable, then it is called almost irresolvable. Let \(I\) be the set of all subspaces of \(X\), each of which has empty interior. The author proves that \(I\) is a \(\sigma\)-complete ideal on \(X\) iff every open subspace of \(X\) is almost irresolvable (Theorem 2.5). Using this connection between irresolvability and existence of ideals, the author proves that (1) If the continuum is less than the first weakly inaccessible cardinal, then every Hausdorff crowded (= no isolated points) ccc space is almost resolvable (Theorem 2.19); (2) Every crowded ccc space of cardinality less than the first weakly inaccessible cardinal is almost resolvable (Theorem 2.22); (3) There is a measurable cardinal iff there is a crowded almost irresolvable space (Theorem 3.20).