A note on spaces that are finitely an \(F\)-space. (Q260589)

We recall that a Tychonoff space \(X\) is finitely an \(F\)-space (a concept first considered by \textit{M. Henriksen} and \textit{R. Wilson} in 1992 [Topology Appl. 44, No. 1-3, 175-180 (1992; Zbl 0801.54014)]) if \(\beta X\) is expressible as a union of finitely many closed \(F\)-spaces. This article shows that the property of being finitely an \(F\)-space can be characterized in terms of algebraic properties of the ring \(C(X)\), namely: a Tychonoff space \(X\) is finitely an \(F\)-space if and only if \(C(X)\) is finitely 1-convex. This proposition extends a similar result, for the case when \(X\) is a normal space, by \textit{S. Larson} [Ann. Fac. Sci. Toulouse, Math. (6) 19, Spec. Issue, 111-141 (2010; Zbl 1220.06008)].NEWLINENEWLINE The result is shown in the more general context of frames. First, the authors extend to frames the concept of a finitely \(F\)-space by defining what it means to say a frame is finitely an \(F\)-frame. This is a conservative extension, in the sense that a space \(X\) is finitely an \(F\)-space if and only if its frame \(\mathfrak OX\) of open sets is finitely an \(F\)-frame. Then they prove that a completely regular frame \(L\) is finitely an \(F\)-frame if and only if the ring \(\mathcal RL\) of real functions on \(L\) is finitely 1-convex. The result for spaces follows as an immediate corollary, since the rings \(C(X)\) and \(\mathcal R(\mathfrak OX)\) are isomorphic for any Tychonoff space \(X\).