2018/04/09 by Juhász, István, Soukup, Lajos, Szentmiklóssy, Zoltán
#03E35 #54A25 #54A35 #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.1804.03019
Given a topological property P, we say that the space X is P-generated if for any subset A⊂ X that is not open in X there is a subspace Y ⊂ X with property P such that A∩ Y is not open in Y. (Of course, in this definition we could replace "open" with "closed".) In this paper we prove the following two results: (1) Every Lindelöf-generated regular space X satisfying |X|=Δ(X)=ω1 is ω1-resolvable. (2) Any (countable extent)-generated regular space X satisfying Δ(X)>ω is ω-resolvable. These are significant strengthenings of our earlier results from [JSSz] which can be obtained from (1) and (2) by simply omitting the "-generated" part. Moreover, the second result improves a recent result of Filatova and Osipov from [FO] which states that Lindelöf-generated regular spaces of uncountable dispersion character are 2-resolvable. [FO] Maria A. Filatova, Alexander V. Osipov On resolvability of Lindelöf generated spaces, arxiv:1712.00803. Siberian Electronic Mathematical Reports, Vol. 14, (2017) pp. 1444-1444. [JSSz] Juhász, István; Soukup, Lajos; Szentmiklóssy, Zoltán, Regular spaces of small extent are ω-resolvable. Fund. Math. 228 (2015), no. 1, 27-46.