Wallman-type compactifications and products

Abstract

Y is a Wallman-type compactification (O. Frink, Amer. J. Math. 86 (1964), 602-607) of X in case there is a normal base Z for the closed sets of X such that the ultrafilter space from Z, denoted ω ( Z ) \omega (Z) , is topologically Y. It is not known if every compactification is Wallman-type. For Z α {Z_\alpha } a normal base for the closed sets of X α {X_\alpha } for each a belonging to an index set Δ \Delta it is shown that the Tychonoff product space ∏ α ∈ Δ ω ( Z α ) {\prod _{\alpha \in \Delta }}\omega ({Z_\alpha }) is a Wallman compactification of ∏ α ∈ Δ X α {\prod _{\alpha \in \Delta }}{X_\alpha } . Also for X ⊂ T ⊂ ω ( Z ) X \subset T \subset \omega (Z) with Z a normal base for the closed sets of X, a proof that ω ( Z ) \omega (Z) is a Wallman-type compactification of T is indicated.