Let $\omega_1$ have the discrete topology. Its Fortissimo space is $X=\omega_1\cup\{\infty\}$ where neighborhoods of $\infty$ are co-countable.

A space is radial provided for every subset $A$ and point in its closure, there exists a transfinite sequence within $A$ converging to it (see d-4 Pseudoradial spaces).

Then $\infty\in cl(A)$ for any uncountable $A$, and given the "transfinite sequence" $a_\alpha=\min\{a\in A:\beta<\alpha\Rightarrow a>a_\beta\}$ for $\alpha<\omega_1$, we have that every neighborhood of $\infty$ contains a final subsequence. Since $\infty$ is the only possible point added by the closure operator, this shows $X$ is radial.

The problem is: is $a_\alpha$ actually a transfinite sequence? This mapping from the ordinal space $\omega_1$ to $X$ is not continuous as its image is discrete. In fact, every convergent continuous image of an infinite limit ordinal in $X$ is eventually constant. Under this interpretation, the space is not radial.

After more careful review of my reference and JDH's comment, I think it's clear that the first interpretation is correct, and the space is in fact radial.

However, does the more restrictive concept appear in the literature anywhere?

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