The proof of the interpolation property of the way-below relation of a continuous poset

(Note)

Martín Escardó

March 2000

It is fair to say that there is only one proof of the interpolation property of the waybelow relation of a continuous poset. Two versions can be found in [1, pages 46–47] and [2, pages 289–290]. Here is another.

1 If x ≪ y holds in a continuous poset, then for every directed set D with y ≤ ⋁ D there is d ∈ D with x ≪ d. Proof By continuity of the poset, the set I =^{def} ⋃{↓ ↓ d | d ∈ D} has the same join as D. Being a directed union of ideals, I is an ideal. By definition of the way-below relation, we conclude that x ∈ I, which gives the result. □

2 If x ≪ y holds in a continuous poset, then there is an element b with x ≪ b ≪ y. Proof The element y is the directed join of the elements b ≪ y. □

References

[1] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. A Compendium of Continuous Lattices. Springer-Verlag, 1980.

[2] P.T. Johnstone. Stone Spaces. Cambridge University Press, Cambridge, 1982.

Martín H. Escardó School of Computer Science University of Birmingham Birmingham B15 2TT England

m.escardo@cs.bham.ac.uk
www.cs.bham.ac.uk/~mhe/