On monotone stability (Q2437651)

The notion of stable spaces was introduced by \textit{A. V. Arkhangel'skij} [Trans. Mosc. Math. Soc. 1985, 1--22 (1985); translation from Tr. Mosk. Mat. O.-va 47, 3--21 (1984; Zbl 0588.54018)]: a Tychonoff space is stable if for each continuous image \(Y\) of \(X\) the \(i\)-weight of \(Y\) is equal to the net weight of \(Y\). The author of this paper defines monotonically stable spaces, a monotone version of stability. This class of spaces is wide enough. For example, each \(\sigma\)-product (each \(\Sigma\)-product) of Lindelöf \(\Sigma\)-spaces is monotonically stable. Relations with monotone monolithicity introduced by \textit{V. V. Tkachuk} [Topology Appl. 156, No. 4, 840--846 (2009; Zbl 1165.54009)] are investigated. The author proves a function space duality for monotone versions of monolithic and stable properties: the function space \(C_p(X)\) is monotonically monolithic (monotonically stable) if and only if \(X\) is monotonically stable (monotonically monolithic).