Better-quasi-ordering

A quasi-order is better-quasi-ordered if every array into it from every barrier in better-quasi-order theory is good: some barrier members $s◃t$ satisfy $f(s)\le f(t)$. Singleton barriers recover the usual infinite-sequence condition for a well-quasi-ordering. The stronger condition is useful for infinitary closure operations for which well-quasi-ordering alone fails. Equivalently one may use continuous maps from infinite subsets of an infinite subset of the natural numbers, with the discrete topology on the range, and compare $X$ with $X\setminus \{\min X\}$.