Collectionwise Hausdorff spaces

Subjects: Topology
Links: Collectionwise Normal Spaces, Hausdorff Spaces

Def: A topological space $X$ is collectionwise $T_2$ or collectionwise Hausdorff if given any closed discrete subset of $X$ $\{x_{\alpha }\mid \alpha <\kappa \}$ there's a celular family $\{U_{\alpha }\mid \alpha <\kappa \}$ such that $x_{\alpha }\in U_{\alpha }$ for every $\alpha <\kappa$.

Def: A topological space $X$ is strongly collectionwise $T_2$ or strongly collectionwise Hausdorff if given any closed discrete subset of $X$ $\{x_{\alpha }\mid \alpha <\kappa \}$ there's a discrete open family $\{U_{\alpha }\mid \alpha <\kappa \}$ such that $x_{\alpha }\in U_{\alpha }$ for every $\alpha <\kappa$.

Obs: Every collectionwise normal space is strongly collectionwise Hausdorff, and every strongly collectionwise Hausdorff is collectionwise Hausdorff.

Obs: Every collectionwise $T_2$ space is $T_2$.