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Keywords:
Martin's Axiom; Baire spaces; resolvable spaces; $\omega$-resolvable spaces; almost resolvable spaces; almost-$\omega$-resolvable spaces; infinite $\pi$-network
Summary:
We prove that, assuming MA, every crowded $T_0$ space $X$ is $\omega$-resolvable if it satisfies one of the following properties: (1) it contains a $\pi$-network of cardinality $< \frak{c}$ constituted by infinite sets, (2) $\chi(X) < \frak{c}$, (3) $X$ is a $T_2$ Baire space and $c(X) \leq \aleph_0$ and (4) $X$ is a $T_1$ Baire space and has a network $\Cal{N}$ with cardinality $< \frak{c}$ and such that the collection of the finite elements in it constitutes a $\sigma$-locally finite family. Furthermore, we prove that the existence of a $T_1$ Baire irresolvable space is equivalent to the existence of a $T_1$ Baire $\omega$-irresolvable space, and each of these statements is equivalent to the existence of a $T_1$ almost-$\omega$-irresolvable space. Finally, we prove that the minimum cardinality of a $\pi$-network with infinite elements of a space $\operatorname{Seq}(u_t)$ is strictly greater than $\aleph_0$.
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