ultracompact

Definition 1

A space X is called ultracompact if it is p-compact for every free ultrafilter on ω (the set of natural numbers).

Definition 2

A topological space X is ultracompact iff every subset family N with the properties:

1. for any subset A in X, we can pick some R in N such that A or the complement of A is contained in R
2. the interiors of members of N cover X has a finite subfamily which covers X.

Reference

Definition 1

A. R. Bernstein, A new kind of compactness for topological spaces, Fund.Math. 66 (1970), 185-193.

Definition 2

D.V.Thampuran, Nets and Compactness, Portugaliae Mathematica Vol.28(1) pp.37-54.