The dimension of closed sets in the Stone-Čech compactification

Abstract:

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