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On cardinal sequences of length less than omega3
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We prove the following consistency result for cardinal sequences of length $<\omega_3$ : if GCH holds and $\lambda \geq \omega_2$ is a regular cardinal, then in some cardinal-preserving generic extension $2^\omega=\lambda$ and for every ordinal $\eta<\omega_3$ and every sequence $f=\left\langle\kappa_\alpha: \alpha<\eta\right\rangle$ of infinite cardinals with $\kappa_\alpha \leq \lambda$ for $\alpha<\eta$ and $\kappa_\alpha=\omega$ if $\operatorname{cf}(\alpha)=\omega_2$, we have that $f$ is the cardinal sequence of some LCS space. Also, we prove that for every specific uncountable cardinal $\lambda$ it is relatively consistent with ZFC that for every $\alpha, \beta<\omega_3$ with $\operatorname{cf}(\alpha)<\omega_2$ there is an LCS space $Z$ such that $\left.\operatorname{CS}(Z)=\langle\omega\rangle_\alpha \gamma \lambda\right\rangle_\beta$.
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MARTÍNEZ ALONSO, Juan Carlos and SOUKUP, Lajos. On cardinal sequences of length less than omega3. Topology and its Applications. 2019. Vol. 260, num. 116-125. ISSN 0166-8641. [consulted: 18 of October of 2026]. Available at: https://hdl.handle.net/2445/193559