On monotone stability

On some monotone properties

“…The space S being Lindelöf is realcompact, so it can be embedded as a closed subspace in R T for some set T . Besides, S is not stable because ω = iw(S) < nw(S) = c. Since any monotonically stable space is stable (see [17,Corollary 3.3]), we conclude that S is not monotonically stable.…”

“…It was proved in [17,Corollary 3.12] that X is monotonically stable if and only if C p (C p (X)) is monotonically stable. Thus we obtain:…”

“…It follows from [1, Corollary 2.15] that C p (X) is monotonically ω-monolithic. Then we can apply [17,Remark 3.13] to see that X is monotonically ω-stable. So the result follows from Theorem 4.5.…”

“…If X is monotonically Sokolov and monotonically ω-stable then by [19,Theorem 4.17] and [17,Remark 3.13] the space C p (X) is monotonically retractable and monotonically ω-monolithic. So we can apply Theorem 5.1 to see that C p (X) is monotonically Sokolov.…”

“…In fact, Arhangel'skii proved that monolithicity and stability are properties two-way dual with respect to the C p -functor, i.e., a topological space X is monolithic if and only if C p (X) is a stable space, and X is a stable space if and only if C p (X) is a monolithic space. Motivated for this fact, R. Rojas-Hernández introduced in [17] the concept of monotonically stable Fix a point a * in X = {X t : t ∈ T }. For x ∈ X, define the support supp(x) of x as the set {t ∈ T : x(t) = a * (t)}.…”