This paper unifies and extends
various theorems which deal with the relationship between the cardinality of discrete
collections and the cardinality of open coverings. For this purpose, the class of spaces
which are irreducible of order α is defined. This class includes the δ𝜃-refinable and
the [α,∞)-refinable spaces. Some examples of applications of this class are: a space is
[α,∞)-compact if and only if it is irreducible of order α and has the α − BW
property and if X is irreducible of order Δ(X), then Δ(X) ≦ L(X) ≦ Δ(X)+.
The open ordinal space [0,Ω) serves as a model to generate examples of
spaces which are irreducible of order α but not irreducible of order β, if
β > α.
