Very I-favorable spaces

We prove that a Hausdorff space $X$ is very $\mathrm I$-favorable if and only if $X$ is the almost limit space of a $σ$-complete inverse system consisting of (not necessarily Hausdorff) second countable spaces and surjective d-open bonding maps. It is also shown that the class of Tychonoff very $\mathrm I$-favorable spaces with respect to the co-zero sets coincides with the d-openly generated spaces.

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