TITLE: Understanding a definition of radial
QUESTION [6 upvotes]: A space $X$ is called radial if, for any $A \subset X$ and any $x \in cl(A)$, there is a transfinite sequence $s=\{a_\alpha: \alpha \in \kappa\} \subset A$ which converges to $x$. What's meaning of "transfinite sequence" here?
Added: If I may ask more, What is difference between redial space and Frechet space? It seems that they are same.

REPLY [2 votes]: As Paul VanKoughnett has remarked in his answer, a transfinite sequence will be any sequence/function $\langle x_\xi \rangle_{\xi < \alpha}$ where $\alpha$ is an ordinal.  Recall that ordinals are topological spaces under the usual order topology, and in essence if $\langle x_\xi \rangle_{\xi < \alpha}$ converges to some point $x$, this is equivalent to the $(\alpha+1)$-sequence obtained by appending $x$ to the end of $\langle x_\xi \rangle_{\xi < \alpha}$ being continuous at $\alpha$.  (Of course, the $\alpha$-sequence may fail to be continuous at limit ordinals $< \alpha$.)
I assume that the prototypical example of a radial non-Fréchet space would be the ordinal space $\omega_1 + 1$.  Here $\omega_1 \in \overline{ [ 0 , \omega_1 ) }$, however there is no ($\omega$-)sequence in $[0 , \omega_1 )$ converging to $\omega_1$ (since $\omega_1$ has uncountable cofinality).  However there is a perfectly good $\omega_1$-sequence converging to $\omega_1$: $x_\xi = \xi$ for all $\xi < \omega_1$.