  The "weakly Hausdorff" property for pseudoradial spaces fails to be naturally
characterized by unique convergence of transfinite sequences. In response, we
develop the category $\mathbf{SPsRad}$ of strongly pseudoradial spaces,
compactly generated spaces whose closed sets are determined by globally
continuous maps from well-ordered spaces. Categorically, $\mathbf{SPsRad}$ is
the coreflective hull of the class of well-ordered spaces, and
$\mathbf{SPsRad}$ is Cartesian closed. The strongly pseudoradial weakly
Hausdorff spaces admit a natural characterization involving unique extensions
of injective maps of well-ordered spaces. We also obtain analogs in
$\mathbf{SPsRad}$ of the fact that for sequential spaces, sequential
compactness is equivalent to countable compactness.
