Compact-range Vietoris powers: from $\omega+1$ to all countable ordinals above $\omega$

First, On the Vietoris Power of a Convergent Sequence settles the highlighted case $\alpha=\omega+1$: the space is second-countable and Lindelöf, but not $\sigma$-compact. Compact families of surjections have finite bounds on where each isolated value occurs; a delayed diagonal defeats every countable compact cover. The stronger manuscript proves, for every $\omega<\alpha<\omega_1$, second countability and Lindelöfness, together with failure of both $\sigma$-compactness and the Menger property. It combines a countable-base argument, transfer through a closed copy of $[0,\omega]$, and a direct fixed-compact-image diagonal. Explicit increasing open covers and the same diagonal defeat all finite selections, proving non-Mengerness. Thus the original question has a positive answer for Lindelöfness and a negative answer for $\sigma$-compactness. “Proved” refers to this classification theorem.

First, On the Vietoris Power of a Convergent Sequence settles the highlighted case $\alpha=\omega+1$: the space is second-countable and Lindelöf, but not $\sigma$-compact. Compact families of surjections have finite bounds on where each isolated value occurs; a delayed diagonal defeats every countable compact cover. The stronger manuscript proves, for every $\omega<\alpha<\omega_1$, second countability and Lindelöfness, together with failure of both $\sigma$-compactness and the Menger property. It combines a countable-base argument, transfer through a closed copy of $[0,\omega]$, and a direct fixed-compact-image diagonal. Explicit increasing open covers and the same diagonal defeat all finite selections, proving non-Mengerness. Thus the original question has a positive answer for Lindelöfness and a negative answer for $\sigma$-compactness. “Proved” refers to this classification theorem.