THE SPACE OF ORDER-PRESERVING FUNCTIONALS AND Π-COMPLETENESS
DOI:
https://doi.org/10.56143/Keywords:
Π-complete space, perfect compactification, order-preserving functional, finite-component coverAbstract
For a Tychonoff space X, let Of(X) denote the space of order-preserving functionals. We construct an open cover of Of(X) induced by an open cover of X and prove that the disjointness of the original cover implies the disjointness of the induced cover. We further establish that Of(βX), the space of order-preserving functionals on the Stone–Čech compactification βX, is a perfect compactification of Of(X). Finally, using the criterion for Pi-completeness, we prove that Of(X) is Pi-complete if and only if X is Pi-complete.
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