The dimension of closed sets in the Stone-Čech compactification

Abstract

In this paper properties of compacta K K in β X ∖ X \beta X\backslash X are studied for Lindelöf spaces X X . If dim K = ∞ {\operatorname {dim}}\,K = \infty , then there is a mapping f : K → T c f:K \to {T^c} such that f f is onto and every mapping homotopic to f f is onto. This implies that there is an essential family for K K consisting of c c disjoint pairs of closed sets. It also implies that if K = ∪ { K α | α > c } K = \cup \left \{ {{K_\alpha }|\alpha > c} \right \} with each K α {K_\alpha } closed, then there is a β \beta such that dim K β = ∞ {\operatorname {dim}}\,{K_\beta } = \infty . Assume K K is a compactum in β X ∖ X \beta X\backslash X as above. Then if dim K = n {\operatorname {dim}}\,K = n , there is a closed set K ′ K’ in K K such that dim K ′ = n {\operatorname {dim}}\,K’ = n and such that every nonempty G δ {G_\delta } -set in K ′ K’ contains an n n -dimensional compactum. This holds for n n finite or infinite. If dim K = n {\operatorname {dim}}\,K = n and K = ∪ { K α | α > ω 1 } K = \cup \left \{ {{K_\alpha }|\alpha > {\omega _1}} \right \} with each K α {K_\alpha } closed, then there must be a β \beta such that dim K β = n {\operatorname {dim}}\,{K_\beta } = n .