*Definition 1 [#ib356752]
A space X is called ultracompact if it is [[p-compact]] for every free ultrafilter on ω (the set of natural numbers).
*Definition 2 [#n911f6d2]
A topological space X is ultracompact iff every subset family N with the properties:
+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
+the interiors of members of N cover X has a finite subfamily which covers X.
*Reference [#zf504a65]
: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.